This page is an introduction to basic topological homotopy theory. We introduce the concept of homotopy between continuous functions and the induced concept of homotopy equivalence of topological spaces. Homotopy classes of paths form the fundamental groupoid of a topological space, the first step in extracting combinatorial data in homotopy theory. We use this example to introduce groupoids and their homotopy theory in general and mention that this models the homotopy theory of those topological spaces that are homotopy 1-types. Then we discuss the concept of covering spaces and use groupoids to give a simple proof of the fundamental theorem of covering spaces, which says that these are euivalent to permutation representations of the fundamental groupoid. This is a simple topological version of the general principle of Galois theory and has many applications. As one example application, we use it to prove that the fundamental group of the circle is the integers.
$\,$
main page: Introduction to Topology
previous chapter: Introduction to Topology 1 – Point-set topology
this chapter: Introduction to Topology 2 – Basic Homotopy Theory
$\,$
For introduction to more general and abstract homotopy theory see instead at Introduction to Homotopy Theory.
$\,$
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
$\,$
In order to handle topological spaces, to compute their properties and to distinguish them, it turns out to be useful to consider not just continuous variation within a topological space, i.e. continuous functions between topological spaces, but also continuous deformations of continuous functions themselves. This is the concept of homotopy (def. below), and its study is called homotopy theory. If one regards topological spaces with homotopy classes of continuous functions between them then their nature changes, and one speaks of homotopy types (remark below).
Of particular interest are homotopies between paths in a topological space. If a loop in a topological space is homotopic to the constant loop, this means that it does not “wind around a hole” in the space. Hence the set of homotopy classes of loops in a topological space, which is a group under concatenation of paths, detects crucial information about the global structure of the space, and hence is called the fundamental group of the space (def. ).
This same information turns out to be encoded in “continuously varying sets” over a topological space, hence in “bundles of sets”, called covering spaces (def. below). As one moves around a loop, then the parameterized set comes back to itself up to a bijection called the monodromy of the loop. This encodes an action or permutation representation of the fundamental group. The fundamental theorem of covering spaces (prop. below) says that covering spaces are equivalently characterized by their monodromy representation of the fundamental group. This is an incarnation of the general principle of Galois theory in topological homotopy theory. Sometimes this allows to compute fundamental groups from behaviour of covering spaces, for instance it allows to prove that the fundamental group of the circle is the integers (prop. below).
In order to formulate and prove these statements, it turns out convenient to do away with the arbitrary choice of basepoint that is involved in the definition of fundamental groups, and instead collect all homotopy classes of paths into a single structure, called the fundamental groupoid of a topological space (example below) an example of a generalization of groups to groupoids (discussed below). The fundamental groupoid may be regarded as an algebraic incarnation of the homotopy type presented by a topological space, up to level 1 (the homotopy 1-type).
The algebraic reflection of the full homotopy type of a topological space involves higher dimensional analogs fo the fundamental group called the higher homotopy groups. We close with an outlook on these below.
$\,$
It is clear that for $n \geq 1$ the Euclidean space $\mathbb{R}^n$ or equivalently the open ball $B_0^\circ(1)$ in $\mathbb{R}^n$ is not homeomorphic to the point space $\ast = \mathbb{R}^0$ (simply because there is not even a bijection between the underlying sets). Nevertheless, intuitively the $n$-ball is a “continuous deformation” of the point, obtained as the radius of the $n$-ball tends to zero.
This intuition is made precise by observing that there is a continuous function out of the product topological space (this example) of the open ball with the closed interval
which is given by rescaling:
This continuously interpolates between the open ball and the point, in that for $t = 1$ it restricts to the identity, while for $t = 0$ it restricts to the map constant on the origin.
We may summarize this situation by saying that there is a diagram of continuous functions of the form
Such “continuous deformations” are called homotopies:
In the following we use this terminlogy:
The topological interval is
the closed interval $[0,1] \subset \mathbb{R}^1$ regarded as a topological space in the standard way, as a subspace of the real line with its Euclidean metric topology,
equipped with the continuous functions
$const_0 \;\colon\; \ast \to [0,1]$
$const_1 \;\colon\; \ast \to [0,1]$
which include the point space as the two endpoints, respectively
equipped with the (unique) continuous function
to the point space (which is the terminal object in Top)
regarded, in summary, as a factorization
of the codiagonal on the point space, namely the unique continuous function $\nabla_\ast$ out of the disjoint union space $\ast \sqcup \ast \simeq Disc(\{0,1\})$ (homeomorphic to the discrete topological space on two elements).
(homotopy)
Let $X,Y \in$ Top be two topological spaces and let
be two continuous functions between them.
A (left) homotopy from $f$ to $g$, to be denoted
is a continuous function
out of the product topological space (this example) of $X$ the topological interval (def. ) such that this makes the following diagram in Top commute:
graphics grabbed from J. Tauber here
hence such that
If there is a homotopy $f \Rightarrow g$ (possibly unspecified) we say that $f$ is homotopic to $g$, denoted
(homotopy is an equivalence relation)
Let $X,Y \in$ Top be two topological spaces. Write $Hom_{Top}(X,Y)$ for the set of continuous functions from $X$ to $Y$.
Then the relating of being homotopic (def. ) is an equivalence relation on this set. The correspnding quotient set
is called the set of homotopy classes of continuous functions.
Moreover, this equivalence relation is compatible with composition of continuous functions:
For $X,Y,Z \in$ Top three topological spaces, there is a unique function
such that the following diagram commutes:
To see that the relation is reflexive: A homotopy $f \Rightarrow f$ from a function $f$ to itself is given by the function which is constant on the topological interval:
This is continuous becaue projections out of product topological spaces are continuous, by the universal property of the Cartesian product.
To see that the relation is symmetric: If $\eta \colon f \Rightarrow g$ is a homotopy then
is a homotopy $g \Rightarrow f$. This is continuous because $1-(-)$ is a polynomial function, and polynomials are continuous, and because Cartesian product and composition of continuous functions is again continuous.
Finally to see that the relation is transitive: If $\eta_1 \colon f \Rightarrow g$ and $\eta_2 \colon g \Rightarrow h$ are two composable homotopies, then consider the “$X$-parameterized path concatenation”
To see that this is continuous, observe that $\{ X \times [0,1/2] \subset X, X \times [1/2,1] \subset X \}$ is a cover of $X \times [0,1]$ by closed subsets (in the product topology) and because $\eta_1(-,2(-))$ and $\eta_2(-,2(-)-1)$ are continuous (being composites of Cartesian products of continuous functions) and agree on the intersection $X \times \{1/2\}$. Hence the continuity follows by this example.
Finally to see that homotopy respects composition: Let
be continuous functions, and let
be a homotopy. It is sufficient to show that then there is a homotopy of the form
This is exhibited by the following diagram
Prop. means that homotopy classes of continuous functions are the morphisms in a category whose objects are still the topological spaces.
This category (at least when restricted to spaces that admit the structure of CW-complexes) is called the classical homotopy category, often denoted
Hence for $X,Y$ topological spaces, then
Moreover, sending a continuous function to its homotopy class is a functor
from the ordinary category Top of topological spaces with actual continuous functions between them.
Let $X,Y \in$ Top be two topological spaces.
is called a homotopy equivalence if there exists
a continuous function the other way around,
homotopies (def. ) from the two composites to the respective identity function:
and
We indicate that a continuous function is a homotopy equivalence by writing
If there exists some (possibly unspecified) homotopy equivalence between topological spaces $X$ and $Y$ we write
(homotopy equivalences are the isomorphisms in the homotopy category)
In view of remark a continuous function $f$ is a homotopy equivalence precisely if its image $\kappa(f)$ in the homotopy category is an isomorphism.
As an object of the homotopy category, a topoogical space is often referred to as a (strong) homotopy type. Homotopy types have a different nature than the topological spaces which present them, in that topological spaces that are far from being homeomorphic may still be equivalent as homotopy types.
(homeomorphism is homotopy equivalence)
Every homeomorphism is a homotopy equivalence (def. ).
(homotopy equivalence is equivalence relation)
Being homotopy equivalent is an equivalence relation on the class of topological spaces.
This is immediate from remark by general properties of categories and functors.
But for the record we spell it out. This involves the construction already used in the proof of prop. :
It is clear that the relation it reflexive and symmetric. To see that it is transitive consider continuous functions
and homotopies
We need to produce homotopies of the form
and
Now the diagram
with $\eta$ one of the given homotopies, exhibits a homotopy $(g_1\circ g_2) \circ (f_2 \circ f_1) \Rightarrow g_1 \circ f_1$. Composing this with the given homotopy $g_1 \circ f_1 \Rightarrow id_X$ gives the first of the two homotopies required above. The second one follows by the same construction, just with the lables of the functions exchanged.
(contractible topological space)
A topological space $X$ is called contractible if the unique continuous function to the point space
is a homotopy equivalence (def. ).
(contractible topological spaces are the terminal objects in the homotopy category)
In view of remark , a topological space $X$ is contractible (def. ) precisely if its image $\kappa(X)$ in the classical homotopy category is a terminal object.
(closed ball and Euclidean space are contractible)
Let $B^n \subset \mathbb{R}^n$ be the unit open ball or closed ball in Euclidean space. This is contractible (def. ):
The homotopy inverse function is necessarily constant on a point, we may just as well choose it to go pick the origin:
For one way of composing these functions we have the equality
with the identity function. This is a homotopy by prop. .
The other composite is
Hence we need to produce a homotopy
This is given by the function
where on the right we use the multiplication with respect to the standard real vector space structure in $\mathbb{R}^n$.
Since the open ball is homeomorphic to the whole Cartesian space $\mathbb{R}^n$ (this example) it follows with example and example that also $\mathbb{R}^n$ is a contractible topological space:
In direct generalization of the construction in example one finds further examples as follows:
The following three graphs
(i.e. the evident topological subspaces of the plane $\mathbb{R}^2$ that these pictures indicate) are not homeomorphic. But they are homotopy equivalent, in fact they are each homotopy equivalent to the disk with two points removed, by the homotopies indicated by the following pictures:
graphics grabbed from Hatcher
$\,$
Let $X$ be a topological space and let
be two paths in $X$, i.e. two continuous functions from the closed interval to $X$, such that their endpoints agree:
Then a homotopy relative boundary from $\gamma_1$ to $\gamma_2$ is a homotopy (def. )
such that it does not move the endpoints:
(homotopy relative boundary is equivalence relation on sets of paths)
Let $X$ be a topological space and let $x, y \in X$ be two points. Write
for the set of paths $\gamma$ in $X$ with $\gamma(0) = x$ and $\gamma(1) = y$.
Then homotopy relative boundary (def. ) is an equivalence relation on $P_{x,y}X$.
The corresponding set of equivalence classes is denoted
Recall the operations on paths: path concatenation $\gamma_2 \cdot \gamma_1$, path reversion $\overline{\gamma}$ and constant paths
(concatenation of homotopy relative boundary-classes of paths)
For $X$ a topological space, then the operation of path concatenation descends to homotopy relative boundary equivalence classes, so that for all $x,y, z \in X$ there is a function
Moreover,
this composition operation is associative in that for all $x,y,z,w \in X$ and $[\gamma_1] \in Hom_{\Pi_1(X)}(x,y)$, $[\gamma_2] \in Hom_{\Pi_1(X)}(y,z)$ and $[\gamma_3] \in Hom_{\Pi_1(X)})(z,w)$ then
this composition operation is unital with neutral elements the constant paths in that for all $x,y \in X$ and $[\gamma] \in Hom_{\Pi_1(X)}(x,y)$ we have
this composition operation has inverse elements given by path reversal in that for all $x, y \in X$ and $[\gamma] \in Hom_{\Pi_1(X)}(x,y)$ we have
(fundamental groupoid and fundamental groups)
Let $X$ be a topological space. Then set of points of $X$ together with the sets $Hom_{\Pi_1(X)}(x,y)$ of homotopy relative boundary-classes of paths (def. ) for all points of points and equipped with the concatenation operation from prop. is called the fundamental groupoid of $X$, denoted
Given a choice of point $x \in X$, then one writes
Prop. says that under concatenation of paths, this set is a group. As such it is called the fundamental group of $X$ at $x$.
The following picture indicates the four non-equivalent non-trivial generators of the fundamental group of the oriented surface of genus 2:
graphics grabbed from Lawson 03
(fundamental group of Euclidean space)
For $n \in \mathbb{N}$ and $x \in \mathbb{R}^n$ any point in the $n$-dimensional Euclidean space (regarded with its metric topology) we have that the fundamental group (def. ) at that point is trivial:
(basepoints)
Definition intentionally offers two variants of the defintion.
The first, the fundamental groupoid is canonically given, without choosing a basepoint. As a result, it is a structure that is not quite a group but, slightly more generally, a “groupoid” (a “group with many objects”). We discuss the concept of groupoids below.
The second, the fundamental group, is a genuine group, but its definition requires picking a base point $x \in X$.
In this context it is useful to say that
a pointed topological space $(X,x)$ is
a topological space $X$;
a $x \in X$ in the underlying set.
a homomorphism of pointed topological spaces $f \;\colon\; (X,x) \longrightarrow (Y,y)$ is a base-point preserving continuous function, namely
a continuous function $f \;\colon\; X \longrightarrow Y$
such that $f(x) = y$.
Hence there is a category, to be denoted, $Top^{\ast/}$, whose objects are the pointed topological spaces, and whose morphisms are tbe base-point preserving continuous functions.
Similarly, a homotopy between morphisms $f, f' \colon (X,x) \to (Y,y)$ in $Top^{\ast/}$ is a homotopy $\eta \colon f \Rightarrow f'$ of underlying continuous functions, as in def. , such that the corresponding function
preserves the basepoints in that
These pointed homotopies still form an equivalence relation as in prop. and hence quotienting these out yields the pointed analogue of the homotopy category from def. , now denoted
In general it is hard to explicitly compute the fundamental group of a topological space. But often it is already useful to know if two spaces have the same fundamental group or not:
(pushforward of elements of fundamental groups)
Let $(X,x)$ and $(Y,y)$ be pointed topological space (remark ) and let
be a continuous function which respects the chosen points, in that $f(x) = y$.
Then there is an induced homomorphism of fundamental groups (def. )
given by sending a closed path $\gamma \colon [0,1] \to X$ to the composite
(fundamental group is functor on pointed topological spaces)
The pushforward operation in def. is functorial, now on the category $Top^{\ast/}$ of pointed topological spaces (remark )
(fundamental group depends only on homotopy classes)
Let $X,Y \in Top^{\ast/}$ be pointed topological space and let $f_1, f_2 \;\colon\; X \longrightarrow Y$ be two base-point preserving continuous functions. If there is a pointed homotopy (def. , remark )
then the induced homomorphisms on fundamental groups (def. ) agree
In particular if $f \;\colon; X \longrightarrow Y$ is a homotopy equivalence (def. ) then $f_\ast \;\colon\; \pi_1(X,x) \to \pi_1(Y,y)$ is an isomorphism.
This follows by the fact that homotopy respects composition (prop. ):
If $\gamma \;\colon\; [0,1] \longrightarrow X$ is a closed path representing a given element of $\pi_1(X,x)$, then the homotopy $f_1 \Rightarrow f_2$ induces a homotopy
and therefore these represent the same elements in $\pi_1(Y,y)$.
If follows that if $f$ is a homotopy equivalence with homotopy inverse $g$, then $g_\ast \colon \pi_1(Y,y) \to \pi_1(X,x)$ is an inverse morphism to $f_\ast \colon \pi_1(X,x) \to \pi_1(Y,y)$ and hence $f_\ast$ is an isomorphism.
Prop. says that the fundamental group functor from def. and remark factors through the classical pointed homotopy category from remark :
(simply connected topological space)
A topological space $X$ for which
$\pi_0(X) \simeq \ast$ (path connected)
$\pi_1(X,x) \simeq 1$ (the fundamental group is trivial, def. ),
is called simply connected.
We will need also the following local version:
(semi-locally simply connected topological space)
A topological space $X$ is called semi-locally simply connected if every point $x \in X$ has a neighbourhood $U_x \subset X$ such that every loop in $X$ is contractible as a loop in $X$, hence such that the induced morphism of fundamental groups (def. )
is trivial (i.e. sends everything to the neutral element).
If every $x$ has a neighbourhood $U_x$ which is itself simyply connected, then $X$ is called a locally simply connected topological space. This implies semi-local simply-connectedness.
(Euclidean space is simply connected)
For $n \in \mathbb{N}$, then the Euclidean space $\mathbb{R}^n$ is a simply connected topological space (def. ).
In def. we extracted the fundamental group at some point $x \in X$ from a larger algebraic structure, that incorporates all the basepoints, to be called the fundamental groupoid. This larger algebraic structure of groupoids is usefully made explicit for the formulation and proof of the fundamental theorem of covering spaces (theorem below) and the development of homotopy theory in general.
Where a group may be thought of as a group of symmetry transformations that isomorphically relates one object to itself (the symmetries of one object, such as the isometries of a polyhedron) a groupoid is a collection of symmetry transformations acting between possibly more than one object.
Hence a groupoid consists of a set of objects $x, y, z, \cdots$ and for each pair of objects $(x,y)$ there is a set of transformations, usually denoted by arrows
which may be composed if they are composable (i.e. if the first ends where the second starts)
such that this composition is associative and such that for each object $x$ there is identity transformation $x \overset{id_X}{\longrightarrow} x$ in that this is a neutral element for the composition of transformations, whenever defined.
So far this structure is what is called a small category. What makes this a (small) groupoid is that all these transformations are to be “symmetries” in that they are invertible morphisms meaning that for each transformation $x \overset{f}{\longrightarrow} y$ there is a transformation the other way around $y \overset{f^{-1}}{\longrightarrow} x$ such that
If there is only a single object $x$, then this definition reduces to that of a group, and in this sense groupoids are “groups with many objects”. Conversely, given any groupoid $\mathcal{G}$ and a choice of one of its objects $x$, then the subcollection of transformations from and to $x$ is a group, sometimes called the automorphism group $Aut_{\mathcal{G}}(x)$ of $x$ in $\mathcal{G}$.
Just as for groups, the “transformations” above need not necessarily be given by concrete transformations (say by bijections between objects which are sets). Just as for groups, such a concrete realization is always possible, but is an extra choice (called a representation of the groupoid). Generally one calls these “transformations” morphisms: $x \overset{f}{\longrightarrow} y$ is a morphism with “source” $x$ and “domain” $y$.
An archetypical example of a groupoid is the fundamental groupoid $\Pi_1(X)$ of a topological space (def. below, for introduction see here): For $X$ a topological space, this is the groupoid whose
objects are the points $x \in X$;
morphisms$x \overset{[\gamma]}{\longrightarrow} y$ are the homotopy relative boundary-equivalence classes $[\gamma]$ of paths $\gamma \colon [0,1] \to X$ in $X$, with $\gamma(0) = x$ and $\gamma(1) = y$;
and composition is given, on representatives, by concatenation of paths. Here the class of the reverse path $\bar\gamma \;\colon\; t \mapsto \gamma(1-t)$ constitutes the inverse morphism, making this a groupoid.
If one chooses a point $x \in X$, then the corresponding group at that point is the fundamental group $\pi_1(X,x) \coloneqq Aut_{\Pi_1(X)}(x)$ of $X$ at that point.
This highlights one of the reasons for being interested in groupoids over groups: Sometimes this allows to avoid unnatural ad-hoc choices and it serves to streamline and simplify the theory.
A homomorphism between groupoids is the obvious: a function between their underlying objects together with a function between their morphisms which respects source and target objects as well as composition and identity morphisms. If one thinks of the groupoid as a special case of a category, then this is a functor. Between groupoids with only a single object this is the same as a group homomorphism.
For example if $f \;\colon\; X \to Y$ is a continuous function between topological spaces, then postcomposition of paths with this function induces a groupoid homomorphism $f_\ast \;\colon\; \Pi_1(X) \longrightarrow \Pi_1(Y)$ between the fundamental groupoids from above.
Groupoids with groupoid homomorphisms (functors) between them form a category Grp (def. below) which includes the categeory Grp of groups as the full subcategory of the groupoids with a single object. This makes precise how groupoid theory is a genralization of group theory.
However, for groupoids more than for groups one is typically interested in “conjugation actions” on homomorphisms. These are richer for groupoids than for groups, because one may conjugate with a different morphism at each object. If we think of groupoids as special cases of categories, then these “conjugation actions on homomorphisms” are natural transformations between functors.
For examples if $f,g \;\colon\; X \longrightarrow Y$ are two continuous functions between topological spaces, and if $\eta \;\colon\; f \Rightarrow g$ is a homotopy from $f$ to $g$, then the homotopy relative boundary classes of the paths $\eta(x,-) \;\colon\; [0,1] \to Y$ constitute a natural transformation between $f_*, g_\ast \;\colon\; \Pi_1(X) \to \Pi_y(Y)$ in that for all paths $x_1 \overset{[\gamma]}{\longrightarrow} x_2$ in $X$ we have the “conjugation relation”
(groupoid)
A small groupoid $\mathcal{G}$ is
for all pairs of objects $(x,y) \in X \times X$ a set $Hom(x,y)$, to be called the set of morphisms with domain or source $x$ and codomain or target $y$;
for all triples of objects $(x,y,z) \in X \times X \times X$ a function
to be called composition
for all objects $x \in X$ an element
to be called the identity morphism on $x$;
for all pairs $x,y \in Hom(x,y)$ of obects a function
to be called the inverse-assigning function
such that
(associativity) for all quadruples of objects $x_1, x_2, x_3, x_4 \in X$ and all triples of morphisms $f \in Hom(x_1,x_2)$, $g \in Hom(x_2,x_3)$ and $h \in Hom(x_3,x_4)$ an equality
(unitality) for all pairs of objects $x,y \in X$ and all moprhisms $f \in Hom(x,y)$ equalities
(invertibility) for all pairs of objects $x,y \in X$ and every morphism $f \in Hom(x,y)$ equalities
If $\mathcal{G}_1, \mathcal{G}_2$ are two groupoids, then a homomorphism or functor between them, denoted
is
a function $F_0 \;\colon\; X_1 \longrightarrow X_2$ between the respective sets of objects;
for each pair $x,y \in X_1$ of objects a function
between sets of morphisms
such that
(respect for composition) for all triples $x,y,z \in X_1$ and all $f \in Hom(x,y)$ and $g \in Hom(y,z)$ an equality
(respect for identities) for all $x \in X$ an equality
For $\mathcal{G}_1, \mathcal{G}_2$ two groupoids, and for $F,G \;\colon\; \mathcal{G}_1 \to \mathcal{G}_2$ two groupoid homomorphisms/functors, then a conjugation or homotopy or natural transformation (necessarily a natural isomorphism)
is
such that
for all $x,y \in X_1$ and $f \in Hom_{\mathcal{G}_1}(x,y)$ an equality
For $\mathcal{G}_1, \mathcal{G}_2$ two groupoids and $F, G, H \colon \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ three functors between them and $\eta_1 \;\colon\; F \Rihtarrow G$ and $\eta_2 \;\colon\; G \Rightarrow H$ conjugation actions/natural isomorphisms between these, there is the composite
with components the composite of the components
This yields for any two groupoid a hom-groupoid
whose objects are the groupoid homomorphisms / functors, and whose morphisms are the conjugation actions / natural transformations.
The archetypical example of a groupoid we already encountered above:
For $X$ a topological space, then its fundamental groupoid (as in def. ) has as set of objects the underlying set of $X$, and for $x,y \in X$ two points, the set of homomorphisms is the set of paths from $x$ to $y4 modulo :
and is given by .
( are special cases of )
A (def. ) is equivalently a in which all are .
While therefore groupoid theory may be regarded as a special case of , it is noteworthy that the two theories are quite different in character. For example theory is which is rich but quite tractable, for instance via tools such as or , while is intricate and becomes tractable mostly by making recourse to higher groupoid theory in the guise of and .
(groupoid of a )
For $\mathcal{C}$ any () , then there is a maximal groupoid inside
sometimes called the of $\mathcal{C}$. This is obtained from $\mathcal{C}$ simply by discarding all those that are not .
For instance
For $\mathcal{C} =$ then $Core(Set)$ is the goupoid of and between them.
For $\mathcal{C}$ then the of this groupoid (prop. ) is the disjoint union of deloopings (example ) of all the :
For $\mathcal{C} =$ then $Core(Vect)$ is the groupoid of and between them.
For $\mathcal{C} =$ then the of this groupoid is the disjoint union of delooping of all the
()
For $X$ any set, there is the $Disc(X)$, whose set of objects is $X$ and whose only morphisms are .
(/ of )
Let $\{\mathcal{G}_i\}_{i \in I}$ be a of . Then their () is the groupoid
whose set of objects is the disjoint union of the sets of objects of the summand groupoids, and whose sets of morphisms between two objects is that of $\mathcal{G}_i$ if both objects are form this groupoid, and is otherwise.
( of )
Let $\{\mathcal{G}_i\}_{i \in I}$ be a of . Their is the [groupoid]]
whose set of objects is the of the sets of objects of the factor groupoids
and whose set of between $(x_i)_{i \in I}$ and $(y_i)_{i \in I}$ is the corresponding Cartesian product of morphisms, with elements denoted
For instance if each of the groupoids is the $\mathcal{G}_i = B G_i$ of a $G_i$ (example ) then the product groupoid is the delooping groupoid of the :
As another example, if $\underset{i \in I}{\sqcup} \mathcal{G}_i$ is the groupoid from example , and if $\mathcal{G}$ is any groupoid, then a groupoid homomorphism of the form
is equivalently a $(f_i)_{i \in I}$ of groupoid homomorphisms
The analogous statement holds for homotopies between groupoid homomorphisms, and so one find that the out of a coproduct of groupoids is the product groupoid of the separate hom-groupoids:
( of )
From def. we see that there is a whose
are the small groupoids;
are the groupoid homomorphisms ().
But since this does not reflect the existence of / between homomorphsims/ of groupoids (def. ) this is not what one is interested in when considering /.
In order to obtain the right notion of category of groupoids that does reflect homotopies, we first consider now the horizontal composition of homotopies/natural transformations.
( of with )
Let $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$, $\mathcal{G}_4$ be groupoid and let
be morphisms and a homotopy $\eta$. Then there is a homotopy
between the respective composites, with components given by
This operation constitutes a groupoid homomorphism/functor
The respect for identities is clear. To see the respect for composition, let
be two composable homotopies. We need to show that
Now for $x$ any object of $\mathcal{G}_1$ we find
Here all steps are unwinding of the definition of horizontal and of ordinary (vbertical) composition of homotopies, except the third equality, which is the functoriality of $F_2$.
( of )
Consider a of , groupoid homomorphsims (functors) and homotopies (natural transformations) as follows:
The of the homotopies to a single homotopy of the form
may be defined in temrs of the horizontal composition of homotopies with morphisms (lemma ) and the (“vertical”) composition of homotopies with themselves, in two different ways, namely by decomposing the above diagram as
or as
In the first case we get
while in the second case we get
These two definitions coincide.
For $x$ an object of $\mathcal{G}_1$, then we need that the following square in $\mathcal{G}_3$
But the ommutativity of the square on the right is the defining compatibility condition on the components of $\eta_2$ applied to the morphism $\eta_1(x)$ in $\mathcal{G}_2$.
( with is )
Consider groupoids, homomorphisms and homotopies of the form
Then horizontal composition with the homotopies (lemma ) constitutes a between the functors of horizontal composition with morphisms (lemma )
It first of all follows that the following makes sense
( of groupoids)
There is also the $Ho(Grpd)$ whose
are small groupoids;
are of groupoid homomorphisms modulo homotopy (i.e. modulo ).
This is usually denoted $Ho(Grpd)$.
Of course what the above really means is that, without quotienting out homotopies, groupoids form a , in fact a , in fact an which is enriched over the naive 1-category of groupoids from remark , hece a with .
()
Given two groupoids $\mathcal{G}_1$ and $\mathcal{G}_2$, then a homomorphism
is an it it is an in the $Ho(Grpd)$ (def. ), hence if there exists a homomorphism the other way around
and a homotopy/natural transformations of the form
( of )
If $X$ and $Y$ are and $f \;\colon\; X \longrightarrow Y$ is a between them, then this induces a groupoid homomorphism (functor) between the respective (def. )
given on objects by the underlying function of $f$
and given on the class of a by the evident postcomposition with $f$
This construction clearly respects and and hence is itself a of the form
from the of to the of groupoids.
But more is true: If $f,g \;\colon\; X \longrightarrow Y$ are two and
is a between them, hence a continuous function
such that $\eta(-,0) = f$ and $\eta(-,1) = g$, then this induces a homotopy between the above groupoid homomorphisms (a natural transformation of functors).
This shows that the fundamental groupoid functor in fact descends to
(In fact this means it even extends to a from the of topological spaces, continuous functions, and -classes of left homotopues, to that of groupoids.)
As a direct consequence it follows that if there is a
between , then there is an induced betwee their
Hence the is a of topological spaces. Of course by prop. the fundamental groupoid is equivalent, as a groupoid, to the disjoint union of the of all the of the given topological spaces, one for each , and hence this is equivalently the statement that the set of connected components and the fundamental groups of a topological space are homotopy invariants.
( of a )
Let $G$ be a . Then there is a groupoid, denoted $B G$, with a single object $p$, with morphisms
the elements of $G$, with composition the multiplication in $G$, with identity morphism the in $G$ and with inverse morphisms the inverse elements in $G$.
This is also called the of $G$ (because the of $B G$ at the unique point is the given group: $\Omega B G \simeq G$).
For $G_1, G_2$ two groups, then there is a between $\phi \colon G_1 \to G_2$ and groupoid homomorphisms $G G_1 \to B_ G_2$: the latter are all of the form $B \phi$, with $(B \phi)_0$ uniquely fixed and $(B \phi)_{p,p} = \phi$.
This means that the construction $B(-)$ is a
into from the category of groups to the of .
But beware that this functor is not fully faithful when homotopies of groupoids are taken into acount, because there are in general non-trivial homotopies between morphims of the form
By definition, such a homotopy (natural transformation) $\eta \;\colon\; B \phi_1 \Rightarrow B \phi_2$ is a choice of a single elemet $\eta_p \in H$ such that for all $g \in G$ we have
hence such that
Therefore notably the induced functor
to the of groupoids is not fully faithful.
But since $B G$ is canonically a in groupoids, we may also regard as a functor
to the of . Since groupoid homomorphisms $B G_1 \to B G_2$ necessarily preserve the basepoint, this makes no difference at this point. But as we now pass to the
then also the homotopies are required to preserve the absepoint, and for homotopies between homomorphisms between delooped groups this means, since there only is a single point, that these homotopies are all trivial. Hence regarded this way the functor is a again, hence an onto its . By prop. below this essential image consists precisely of the (pointed) groupoids:
Groups are equivalently pointed connected groupoids.
( of groupoids)
Let $\{G_i\}_{i \in I}$ be a of . Then there is a groupoid $\underset{i \in I}{\sqcup} B G_i$ which is the disjoint union groupoid (example ) of the groupoids $B G_i$ (example ).
Its set of objects is the index set $I$, and
( of a groupoid)
Given a $\mathcal{G}$ with set of objects $X$, then the “there exists a morphism from $x$ to $y$”, i.e.
is clearly an on $X$. The corresponding set of is denoted
and called the set of of $\mathcal{G}$.
()
Given a $\mathcal{G}$ and an object $x$, then under composition the set $Hom_{\mathcal{G}}(x,x)$ forms a . This is called the $Aut_{\mathcal{G}}(x)$ or vertex group or isotropy group of $x$ in $\mathcal{G}$.
For each object $x$ in a groupoid $\mathcal{G}$, there is a canonical groupoid homomorphism
from the delooping groupoid (def. ) of the automorphism group. This takes the unique object of $B Aut_{\mathcal{G}}(x)$ to $x$ and takes every automorphism of $x$ “to itself”, regarded now again as a morphism in $\mathcal{G}$.
( of )
Let $\mathcal{G}_1$ and $\mathcal{G}_2$ be groupoids. Then a morphism (functor)
is called a if
( depends on basepoint only up to )
For $\mathcal{G}$ a groupoid, let $x$ and $y$ be two in the same (def. ). Then there is a group
By assumption, there exists some morphism from $x$ to $y$
The operation of with this morphism
is clearly a group isomorphism as required.
( between of groupoids)
Let $\{G_i\}_{i \in I}$ and $\{H_j\}_{j \in J}$ be sets of and consider a homomorphism ()
between the corresponding disjoint unions of groupoids (example ).
Then the following are equivalent:
The implication 2) $\Rightarrow 1)$ is immediate.
In the other direction, assume that $F$ is an equivalence of groupoids, and let $G$ be an inverse up to natural isomorphism. It is clear that both induces bijections on connected components. To see that both are isomorphisms of automorphisms groups, observe that the conditions for the natural isomorphisms
are in each separate groupoid $B H_j$ of the form
since there is only a single object. But this means $F_{i,i}$ and $F_{j,j}$ are group isomorphisms.
(every is to a of )
Assuming the , then:
For $\mathcal{G}$ any , then there exists a $\{G_i\}_{i \in I}$ of groups and an (def. )
between $\mathcal{G}$ and a of groupoids (example ). This is called a of $\mathcal{G}$.
Concretely, this exists for $I = \pi_0(\mathcal{G})$ the set of of $\mathcal{G}$ (def. ) and for $G_i \coloneqq Aut_{\mathcal{G}}(x)$ the (def. ) of any object $x$ in the given connected component.
Using the we may find a set $\{x_i\}_{i \in \pi_0(\mathcal{G})}$ of objects of $\mathcal{G}$, with $x_i$ being in the $i \in \pi_0(\mathcal{G})$.
This choice induces a functor
which takes each object and morphism “to itself”.
Now using the once more, we choose in each connected component $i \in \pi_0(\mathcal{G})$ and for each object $y$ in that connected component a morphism
Using this we obtain a functor the other way around
which sends each object to its connected component, and which for pairs of objects $y$, $z$ of $\mathcal{G}$ is given by conjugation with the morphisms choosen above:
It is now sufficient to show that there are conjugations/natural isomorphisms
For the first this is immediate, since we even have equality
For the second we observe that choosing
yields a naturality square by the above construction:
( is )
Let $F \;\colon\; \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ be a homomorphism of .
Assuming the then the following are equivalent:
In one direction, if $F$ has an inverse up to natural isomorphism, then this induces by definition a bijection on connected components, and it induces isomorphism on homotopy groups by lemma .
In the other direction, choose equivalences to as in prop. to get a in the of groupoids as follows:
Here $inc_1$ and $inc_2$ are equivalences of groupoids by prop. . Moreover, by assumption that $F$ is a weak homotopy equivalence $\tilde F$ is the union of of deloopings of isomorphisms of groups, and hence has a strict inverse, in particular a homotopy inverse, hence is in particular an euivalence of groupoids.
In conclusion, when regarded as a diagram in the $Ho(Grpd)$ (def. ), the top, bottom and right moprhism of the above diagram are isomorphisms. It follows that also $f$ is an isomorphism in $Ho(Grpd)$. But this means exactly that it is a homotopy equivalence of groupoids, by def. .
A covering space (def. below) is a “continuous of sets” over a topological space, in just the same way as a is a “continuous of vector spaces”.
$\,$
()
Let $X$ be a . A of $X$ is a
such that there exists an $\underset{i}{\sqcup}U_i \to X$, such that restricted to each $U_i$ then $E \to X$ is over $U_i$ to the (this example) of $U_i$ with the (this example) on a $F_i$,
In summary this says that $p \colon E \to X$ is a covering space if there exists a in of the form
For $x \in U_i \subset X$ a point, then the elements in $F_x = F_i$ are called the of the covering at $x$.
Given two covering spaces $p_i \colon E_i \to X$ , then a between them is a $f \colon E_1 \to E_2$ between the total covering spaces, which respects the in that the following
This defines a $Cov(X)$, the over $X$, whose
are the covering spaces over $X$;
are the homomorphisms between these.
(trivial covering space)
For $X$ a and $S$ a with $Disc(S)$ the on that set, then the out of the
is a , called the trivial covering space over $X$ with $Disc(S)$.
If $E \overset{p}{\longrightarrow} X$ is any covering space, then an of covering spaces of the form
is called a trivialization of $E \overset{p}{\to} X$.
It is in this sense that evry coverin space $E$ is, by definition, locally trvializable.
(covering of circle by circle)
Regard the $S^1 = \{ z \in \mathbb{C} \;\vert\; {\vert z\vert} = 1 \}$ as the of elements of unit in the . For $k \in \mathbb{N}$, consider the continuous function
given by taking a complex number to its $k$th power. This may be thought of as the result of “winding the circle $k$ times around itself”. Precisely, for $k \geq 1$ this is a (def. ) with $k$ leaves at each point.
graphics grabbed from Hatcher
$\,$
$\,$
(covering of circle by real line)
Consider the
from the to the , which,
with the circle regarded as the unit circle in the $\mathbb{C}$, is given by
with the circle regarded as the unit circle in $\mathbb{R}^2$, is given by
We may think of this as the result of “winding the line around the circle ad infinitum”. Precisely, this is a (def. ) with the at each point forming the set $\mathbb{Z}$ of .
$\,$
Here are some basic properties of covering spaces:
(covering projections are )
If $p \colon E \to X$ is a covering space projection, then $p$ is an .
By definition of covering space there exists an $\{U_i \subset X\}_{i \in I}$ and $p^{-1}(U_i) \simeq U_i \times Disc(F_i)$ for all $i \in I$. Since the out of a are (this prop.), it follows that $p$ is an open map when restricted to any of the $p^{-1}(U_i)$. But a general open subset $W \subset E$ is the union of its restrictions to these subspaces:
Since images preserve unions (this prop.) it follows that
is a union of open sets, and hence itself open.
( of covering space is and )
Let $E \overset{p}{\to} X$ be a covering space. Consider the
hence (by the discussion at Top - Universal constructions) the of the $E \times E$, as shown on the right. By the of the , there is the
Then the of $E$ under this function is an and a :
First to see that it is an open subset. It is sufficient to show that for any $e \in E$ there exists an open neighbourhood of $(e,e) \in E \times_X E$.
Now by definition of covering spaces, there exists an open neighbourhood $U_{p(e)} \subset X$ of $p(e) \in X$ such that
It follows that $U_{p(e)} \times \{e\} \subset E$ is an open neighbourhood. Hence by the nature of the , $U_{p(e)} \times U_{p(e)} \subset E \times E$ is an open neighbourhood of $(e,e)$ in $E \times E$ and hence by the nature of the the restriction
is an open neighbourhood of $(e,e)$ in $E \times_X E$.
Now to see that the diagonal is closed, hence that the complement $(E \times_X E) \setminus \Delta(E)$ is an open subset, it is sufficient to show that every point $(e_1, e_2)$ with $e_1 \neq e_2$ but $p(e_1) = p(e_2)$ has an open neighbourhood in this complement.
As before, there is an open neighbourhood $U \subset X$ of $p(e_1) = p(e_2)$ over which the cover trivializes, and hence $U \times \{e_1\}, U \times \{e_2\} \subset E$ are open neighbourhoods of $e_1$ and $e_2$, respectively. These are disjoint by the assumption that $e_1 \neq e_2$. As above, this means that the intersection
is an open subset of the complement of the diagonal in the fiber product.
$\,$
If $E \overset{p}{\longrightarrow} X$ is any (possibly a or a ) then a is a continuous function $\sigma \colon X \to E$ which sends each point in the base to a point in the above it, hence which makes this :
We may think of this as “lifting” each point in the base to point in the fibers “through” the projection map $p$. More generally if $Y \hookrightarrow X$ is a subspace, we may consider such lifts only over $Y$
sometimes called a “local section”. But this suggests that for $Y \overset{f}{\longrightarrow} X$ any continuous function, we consider “lifting its image through $p$”
For example if $Y = [0,1]$ is the , then $f \colon [0,1] \to X$ is a in the base space $X$, and a lift through $p$ of this is a path in the total space which “runs above” the given path. Such lifts of paths through covering projections is the topic of below.
Here it is of interest to consider the lifting problem subject to some constraint. For instance we will want to consider lifts of paths $\gamma \colon [0,1] \to X$ through a covering projection, subject to the condition that the starting point $\gamma(0)$ is lifted to a prescribed point $p \in E$.
Since such a point is equivalently a continuous function $const_p \colon \ast \to X$ out of the , this is the same as asking for a continuous function $\sigma$ that makes both triangles in the following :
This is an example of a general situation which plays a central role in : We say that a square
is a and that a diagonal morphism
such that both resulting triangles commute is a . If such a lift exists for for the given $p$ and for each $i$ taken from some of morphisms, then one says that $p$ has the against this class.
We now discuss some satisfied by covering spaces:
A first application of the lifting theorem is that it gives the concept of the (def. , prop. below) which is central to the theory of coverign spaces.
These lifting properties will be used in below for the computation of and higher of some topological spaces.
$\,$
( out of into are unique relative to any point)
Let
$E \overset{p}{\to} X$ be a covering space,
$Y$ a
$f \;\colon\; Y \longrightarrow X$ a .
$\hat f_1, \hat f_2 \;\colon\; Y \longrightarrow E$ two of $f$, in that the following :
for $i \in \{1,2\}$.
If there exists $y \in Y$ such that $\hat f_1(y) = \hat f_2(y)$ then the two lifts already agree everywhere: $\hat f_1 = \hat f_2$.
By the of the
the two lifts determine a single continuous function of the form
Write
for the on $E$ in the fiber product. By lemma this is an open subset and a closed subset of the fiber product space. Hence by continuity of $(\hat f_1, \hat f_2)$ also its pre-image
is both closed and open, hence also its is open in $Y$.
Moreover, the assumption that the functions $\hat f_1$ and $\hat f_2$ agree in at least one point means that the above pre-image is . Therefore the assumption that $Y$ is implies that this pre-image coincides with all of $Y$. This is the statement to be proven.
(path lifting property)
Let $p \colon E \to X$ be any . Given
$\gamma \colon [0,1] \to X$ a in $X$,
$\hat x_0 \in E$ be a lift of its starting point, hence such that $p(\hat x_0) = \gamma(0)$
then there exists a unique path $\hat \gamma \colon [0,1] \to E$ such that
it is a lift of the original path: $p \circ \hat \gamma = \gamma$;
it starts at the given lifted point: $\hat \gamma(0) = \hat x_0$.
In other words, every in of the form
has a unique :
First consider the case that the covering space is trival, hence of the form
By the of the in this case a lift $\hat \gamma \colon [0,1] \to X \times Disc(S)$ is equivalently a of continuous functions
Now the lifting condition explicitly fixes $pr_1(\hat \gamma) = \gamma$. Moreover, a continuous function into a $Disc(S)$ is , and since $[0,1]$ is a this means that $pr_2(\hat \gamma)$ is in fact a (this example), hence uniquely fixed to be $pr_2(\hat \gamma) = \hat x_0$.
This shows the statement for the case of trivial covering spaces.
Now consider any covering space $p \colon E \to X$. By definition of covering spaces, there exists for every point $x \in X$ a $U_x \subset X$ such that the restriction of $E$ to $U_x$ becomes a trivial covering space:
Consider such a choice
This is an of $X$. Accordingly, the
constitute an open cover of the $[0,1]$.
Now the is a , so that this cover has a finite open subcover. By the , each element in this finite subcover is a disjoint union of open intervals. The collection of all these open intervals is an open refinement of the original cover, and by compactness it once more has a finite subcover, now such that each element of the subcover is guaranteed to be a single open interval.
This means that we find a of points
with $t_0 = 0$ and $t_{n+1} = 1$ such that for all $0 \lt j \leq n$ there is $x_j \in X$ such that the corresponding path segment
is contained in $U_{x_j}$ from above.
Now assume that $\hat \gamma\vert_{[0,t_j]}$ has been found. Then by the triviality of the covering space over $U_{x_j}$ and the first argument above, there is a unique lift of $\gamma\vert_{[t_j, t_{j+1}]}$ to a continuous function $\hat \gamma|_{[t_j,t_{j+1}]}$ with starting point $\hat \gamma(t_j)$. Since $[0,t_{j+1}]$ is the $[0,t_j] \underset{\{t_j\}}{\sqcup} [t_j,t_{j+1}]$ (this example), it follows that $\hat \gamma|_{[0,t_j]}$ and $\hat \gamma\vert_{[t_j,t_{j+1}]}$ uniquely glue to a continuous function $\hat \gamma\vert_{[0,t_{j+1}]}$ which lifts $\gamma\vert_{[0,t_{j+1}]}$.
By over $j$, this yields the required lift $\hat \gamma$.
Conversely, given any lift, $\hat \gamma$, then its restrictions $\hat \gamma\vert_{[t_j, t_{j+1}]}$ are uniquely fixed by the above inductive argument. Therefore also the total lift is unique. Altrnatively, uniqueness of the lifts is a special case of lemma .
( of )
Let
$E \overset{p}{\to} X$ be a covering space;
$Y$ a .
Then every of the form
has a unique
It is clear what the lift must be: For every point $y \in Y$ the situation restricts to that of path lifting
And so at each point $y \in Y$ the lift of $\eta(x,-)$ must be the unique path that lifts this with starting point $\hat \hat f(y)$. We just need to see that this lift is a continuous function.
To that end we generalize he proof of the path lifting to connected open neighbourhoods of points in $Y$:
Let $\{U_i \subset X\}_{i \in I}$ be an open cover over which the covering space trivializes. Then the pre-images $\{\eta^{-1}(U_i) \subset Y \times [0,1]\}_{i \in I}$ is an open cover of the product space. By nature of the and the on $[0,1]$, each of the $\eta^{-1}(U_i)$ is a union of Cartesian products $V_j \times I_j$ with $V_i \subset Y$ an open subset of $Y$ and $I_i \subset [0,1]$ an interval. Hence there is an open cover of the form
with the property that for each $j$ there exists $i \in I$ with $\eta(V_j \times I_j) \subset U_i$.
Now by the fact that $[0,1]$ is a , for each $y \in Y$ there exists a finite set $K_y \subset J$ such that
still restricts to a cover of $\{y\} \times [I]$. Since $K$ is finite, the intersection
is still open, and so also
still restricts to a cover of $\{y\} \times [0,1]$.
This means that the same argument as for the path lifting in lemma provides a unique lift $\widehat{\eta\vert_{V_y \times [0,1]}}$ for each $y \in Y$.
Moreover, for $y_1, y_2 \in Y$ two points, these lifts clearly have to agree on $V_{y_1} \cap V_{y_2}$.
Since $\{V_y \times [0,1] \subset Y \times [0,1]\}_{y \in Y}$ is an open cover, means that there is a unique function $\hat \eta$ that restricts to all these local lifts (this prop). This is the required lift.
( are )
Continuous functions that satisfy the , hence that have the against continuous functions of the form $Y \overset{(id, const_0)}{\longrightarrow} Y \times [0,1]$ are called . Hence prop. says that covering projections are in particular Hurewicz fibrations.
( for given lifts of paths relative starting point)
Let $p \colon E \to X$ be a . Then given a relative the starting point between two in $X$,
there is for every lift $\hat \gamma_1, \hat \gamma_2$ of these two paths to paths in $E$ with the same starting point a unique homotopy
between the lifted paths that lifts the given homotopy:
For of the form
there is a unique diagonal in the lower diagram, as shown.
Moreover if the homotopy $\eta$ also fixes the endpoint, then so does the lifted homotopy $\hat \eta$.
There are horizontal such that the following diagram commutes
Let $(E,e) \overset{p}{\longrightarrow} (X,x)$ be a and let $f \colon (Y,y) \longrightarrow (X,x)$ be a point-preserving such that the image of the of $(Y,y)$ is contained within the image of the fundamental group of $(E,e)$ in that of $(X,x)$:
Then for $\ell_Y$ a in $(Y,y)$ that happens to be a , every lift of its image path $f \circ \ell$ in $(X,x)$ to a path $\widehat{f\circ \ell_Y}$ in $(E,e)$ is also a loop there.
By assumption, there is a loop $\ell_E$ in $(E,e)$ and a homotopy fixing the endpoints of the form
Then by the homotopy lifting property as in example , there is a homotopy in $(E,e)$ relative to the basepoint
and lifting the homotopy $\eta_X$. Therefore $\eta_E$ is in fact a homotopy between loops, and so $\widehat{f \circ \ell_Y}$ is indeed a loop.
(lifting theorem)
Let
$p \colon E \to X$ be a ;
$e \in E$ a point, with $x \coloneqq p(e)$ denoting its image,
$Y$ be a and ;
$y \in Y$ a point
$f \colon (Y,y) \longrightarrow (X,x)$ a such that $f(y) = x$.
Then the following are equivalent:
There exists a unique lift $\hat f$ in the diagram
of .
The of the of $Y$ under $f$ in that of $X$ is contained in the image of the fundamental group of $E$ under $p$:
Moreover, if $Y$ is path-connected, then the lift in the first item is unique.
The implication $1) \Rightarrow 2)$ is immediate. We need to show that the second statement already implies the first.
So assume that $f_\ast(\pi_1(Y,y)) \subet p_\ast(\pi_1(E,e))$. If a lift exists, then its uniqueness is given by lemma . Hence we need to exhibit a lift.
Since $Y$ is connected and locally path-connected, it is also a (this prop.). Hence for every point $y' \in Y$ there exists a $\gamma$ connecting $y$ with $y'$ and hence a path $f \circ \gamma$ connecting $x$ with $f(y')$. By the path-lifting property (lemma ) this has a unique lift
Therefore
is a lift of $f(y')$.
We claim now that this pointwise construction is independent of the choice $\gamma$, and that as a function of $y'$ it is indeed continuous. This will prove the claim.
First, by the path lifting lemma the lift $\widehat{\f \circ \gamma}$ is unique given $f \circ \gamma$, and hence $\hat f(y')$ depends at most on the choice of $\gamma$.
Hence let $\gamma' \colon [0,1] \to Y$ be another path in $Y$ that connects $y$ with $y'$. We need to show that then $\widehat{f \circ \gamma'} = \widehat{f \circ \gamma}$.
First observe that if $\gamma'$ is related to $\gamma$ by a , so that then also $f \circ \gamma'$ is related to $f \circ \gamma$ by a homotopy, then this is the statement of the homotopy lifting property in the form of example .
Next write $\bar\gamma'\cdot \gamma$ for the of the path $\gamma$ with the of the path $\gamma'$. This is hence a loop in $Y$, and so $f \circ (\bar\gamma'\cdot \gamma)$ is a loop in $X$. The assumption that $f_\ast(\pi_1(Y,y)) \subset p_\ast(\pi_1(E,e))$ implies, via example , that the path $\widehat{f \circ (\bar \gamma' \cdot \gamma)}$ which lifts this loop to $E$ is itself a loop in $E$.
By uniqueness of path lifting, this means that the lift of
coincides at $1 \in [0,1]$ with that of $f \circ \gamma'$ at 1. But $\gamma' \cdot (\bar \gamma' \cdot \gamma)$ is homotopic (via reparameterization) to just $\gamma$. Hence it follows now with the first statement that the lift of $f \circ \gamma'$ indeed coincides with that of $f \circ \gamma$.
This shows that the above prescription for $\hat f$ is well defined.
It only remains to show that the function $\hat f$ obtained this way is continuous.
Let $y' \in Y$ be a point and $W_{\hat f(y')} \subset E$ an open neighbourhood of its image in $E$. It is sufficient to see that there is an open neighbourhood $V_{y'} \subset Y$ such that $\hat f(V_y) \subset W_{\hat f(y')}$.
Let $U_{f(y')} \subset X$ be an open neighbourhood over which $p$ trivializes. Then the restriction
is an open subset of the product space. Consider its further restriction
to the
which is itself an open subset. Since $p$ is an (this prop.), the subset
is open, hence so is its pre-image
Since $Y$ is assumed to be , there exists a path-connected open neighbourhood
By the uniqueness of path lifting, the image of that under $\hat f$ is
This shows that the lifted function is continuous. Finally that this continuous lift is unique is the statement of lemma .
The lifting theorem implies that there are “universal” covering spaces:
It makes sense to speak of universal covering space, because any two are isomorphic:
Since both $E_1$ and $E_2$ are simply connected, the assumption of the lifting theorem for covering spaces is satisfied (prop. ). This says that there are horizontal continuous function making the following diagrams commute:
and that these are unique once we specify the image of a single point, which we may freely do (in the given fiber).
So if we pick any point $x \in X$ and $\hat x_1 \in E_1$ with $p(\hat x) = x$ and $\hat x_2 \in E_2$ with $p(\hat x_2) = x$ and specify that $f(\hat x_1) = \hat x_2$ and $g(\hat x_2) = \hat x_1$ then uniqueness applied to the composites implies $f \circ g = id_{E_{2}}$ and $g \circ f = id_{E_1}$.
( of the )
The , which is simply connected by example , equipped with the projection from example
Since the lift of a path through a is unique once the lift of the starting point is chosen (lemma ) every path in the base space determines a between the over its endpoints. By the of covering spaces as in example this function only depends on the of the path under . Therefore this fiber-assignment is in fact an of the of the base space on sets, called a (def. below). In particular, associated with any homotopy-class of a , hence of an element in the , there is associated a of the fiber over the loop’s basepoint with itself, hence a of the . This is called the of the covering space. It is a measure for how the coverign space fails to be globally trivial.
In fact the (prop. ) below says that the representation characterizes the covering spaces completely and faithfully. This means that covering spaces may be dealt with completely with tools from and , a fact that we make use of in the computation of examples below.
$\,$
()
Let $\mathcal{G}$ be a . Then:
A of $\mathcal{G}$ is a groupoid homomorphism ()
to the groupoid of the category of (example ). Hence this is
For each object $x$ of $\mathcal{G}$ a $V_x$;
for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a $\rho(f) \;\colon\; V_x \to V_y$
such that
(respect for composition) for all composable morphisms $x \overset{f}{\to}y \overset{g}{\to} z$ in the groupoid we have an
(respect for identities) for each object $x$ of the groupoid we have an equality
Similarly a of $\mathcal{G}$ is a groupoid homomorphism ()
to the groupoid core of . Hence this is
For each object $x$ of $\mathcal{G}$ a $S_x$;
for each morphism $x \overset{f}{\longrightarrow} y$ of $\mathcal{G}$ a $\rho(f) \;\colon\; S_x \to S_y$
such that composition and identities are respected, as above.
For $\rho_1$ and $\rho_2$ two such representations, then a homomorphism of representations
is a between these functors, hence is
for each object $x$ of the groupoid a (linear) function
such that for all morphisms $x \overset{f}{\longrightarrow} y$ we have
By def. the representations of $\mathcal{G}$ in $Core(\mathcal{C})$ and homomorphisms between them constitute a called the
( are of )
If here \mathcal(G) = B G
is the groupoid of a $\mathcal(G$ (example ), then a of $B G$ is a of $G$ (def. ), and one writes
for the .
For each object $x \in X$ the canonical inclusion of the delooping groupoid of the (from def. )
induces by precomposition a homomorphism of representation groupoids:
We say that a groupoid representation is faithful or free if for all objects $x$ its restriction to a group representation of $Azt_{\mathcal{G}}(x)$ this way is transitive or free, respectively.
Here the representation $\rho$ of a group $G$ on some set $S$
if for all pairs of elements $s_1, s_2 \in S$ there is a $g \in G$ such that $\rho(g)(s_1) = s_2$;
if whenever $g(s) = s$ holds for all $s \in S$ then $g$ is the .
( are of )
Assuming the then the following holds:
Let $\mathcal{G}$ be a . Then its $Rep(\mathcal{G})$ (def. ) is (def. ) to the (example ) indexed by the set of $\pi_0(\mathcal{G})$ (def. ) of (example ) of the $G_i \coloneqq Aut_{\mathcal{G}}(x_i)$ (def. ) for $x_i$ any object in the $i$th connected component:
Let $\mathcal{C}$ be the category that the representation is on (e.g. $\mathcal{C} =$ for ). Then by definition
Consider the injection functor of the from lemma
By lemma the pre-composition with this constitutes a functor
and by combining lemma with lemma this is an . Finally, by example the groupoid on the right is the product groupoid as claimed.
( of a )
Let $X$ be a and $E \overset{p}{\to} X$ a (def. ). Write $\Pi_1(X)$ for the of $X$ (example ).
Define a groupoid homomorphism
to the groupoid core of the of (example ), hence a (example ), as follows:
to a point $x \in X$ assign the $p^{-1}(\{x\}) \in Set$;
to the of a $\gamma$ connecting $x \coloneqq \gamma(0)$ with $y \coloneqq \gamma(1)$ in $X$ assign the function $p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\})$ which takes $\hat x \in p^{-1}(\{x\})$ to the endpoint of a path $\hat \gamma$ in $E$ which lifts $\gamma$ through $p$ with starting point $\hat \gamma(0) = \hat x$
This construction is well defined for a given representative $\gamma$ due to the unique path-lifting property of covering spaces (lemma ) and it is independent of the choice of $\gamma$ in the given homotopy class of paths due to the (example ). Similarly, these two lifting properties give that this construction respects composition in $\Pi_1(X)$ and hence is indeed a homomorphism of groupoids (a functor).
(extracting is )
Given a between two $E_i \overset{p_i}{\to} X$, hence a $f \colon E_1 \to E_2$ which respects in that the
, then the component functions
are compatible with the monodromy $Fib_{E}$ (def. ) along any $\gamma$ between points $x$ and $y$ from def. in that the following of
This means that $f$ induces a homotopy () between the monodromy homomorphisms (functors)
of $E_1$ and $E_2$, respectively, and hence that constructing monodromy is itself a homomorphism from the of of $X$ to that of of the of $X$:
Let $e \in p_1^{-1}(x)$ be an element, and $\hat \gamma \colon [0,1] \to E_1$ a lift of $\gamma$ to $E_1$ with $\hat \gamma(0) = e$. This means by definition of monodromy that
Since the function $f$ is compatible with the covering projections, the image $f\circ \hat \gamma$ is a lift of $\gamma$ to $E_2$, with $(f \circ \hat \gamma)(0) = f(\hat \gamma(0)) = f(e)$. Therefore
In conclusion, for each element $e \in p_1^{-1}(x)$ we have
This means that the square commutes, as claimed.
( of covering space)
Let $E \overset{p}{\longrightarrow} X$ be a .
Then the $\Pi_1(E)$ of the total space $E$ is the groupoid
whose
are pairs $(x,\hat x)$ consisting of a point $x \in X$ and en element $\hat x \in Fib_E(x)$;
$[\hat \gamma] \colon (x,\hat x) \to (x', \hat x')$ are morphisms $[\gamma] \colon x \to x'$ in $\Pi_1(X)$ such that $Fib_E([\gamma])(\hat x) = \hat x'$.
This is also called the of the functor $Fib_E \;\colon\; \Pi_1(X) \to Core(Set)$, and denoted
( is if is and )
Let $X$ be a and let $E \overset{p}{\to} X$ be a . Then the total space $E$ is
$\,$
()
Let
$(X,\tau)$ be a ,
$\rho \in Rep(\Pi_1(X),Set)$ a of its .
Consider the of all the sets appearing in this representation
For
$U \subset X$ an
which is
for which every element of the $\pi_1(U,x)$ becomes trivial under $\pi_1(U,x) \to \pi_1(X,x)$,
for $\hat x \in \rho(x)$ with $x \in U$
consider the subset
The collection of these defines a (prop. below). Write $\tau_{\rho}$ for the corresponding topology. Then
is a . It canonically comes with the function
Finally, for
a of permutation representations, there is the evident induced function
The construction $\rho \mapsto E(\rho)$ in def. is well defined and yields a of $X$.
Moreover, the construction $f \mapsto Rec(f)$ yields a homomorphism of covering spaces.
First to see that we indeed have a , we need to check (by this prop.) that every point is contained in some base element, and that every point in the intersection of two base elements has a base neighbourhood that is still contained in that intersection.
So let $x \in X$ be a point. By the assumption that $X$ is there exists an $U_x \subset X$ such that every loop in $U_x$ on $x$ is contractible in $X$. By the assumption that $X$ is a , this contains an open neighbourhood $U'_x \subset U_x$ which is and, as every subset of $U_x$, it still has the property that every loop in $U'_x$ based on $x$ is contractible as a loop in $X$. Now let $\hat x \in E$ be any point over $x$, then it is contained in the base open $V_{U'_x,x}$.
The argument for the base open neighbourhoods contained in intersections is similar.
Then we need to see that $p \colon E(\rho) \to X$ is a . Since taking pre-images preserves unions (this prop.), and since by semi-local simply connectedness and local path connectedness every neighbourhood contains an open neighbourhood $U \subset X$ that labels a base open, it is sufficient to see that $p^{-1}(U)$ is a base open. But by the very assumption on $U$, there is a unique morphism in $\Pi_1(X)$ from any point $x \in U$ to any other point in $U$, so that $\rho$ applied to these paths establishes a bijection of sets
thus exhibiting $p^{-1}(U)$ as a union of base opens.
Finally we need to see that this continuous function $p$ is a covering projection, hence that every point $x \in X$ has a neighbourhood $U$ such that $p^{-1}(U) \simeq U \times \rho(x)$. But this is again the case for those $U$ all whose loops are contractible in $X$, by the above identification via $\rho$, and these exist around every point by semi-local simply-connetedness of $X$.
This shows that $p \colon E(\rho) \to X$ is a covering space. It remains to see that $Rec(f) \colon E(\rho_1) \to E(\rho_2)$ is a homomorphism of covering spaces. Now by construction it is immediate that this is a function over $X$, in that this :
So it only remains to see that $Rec(f)$ is a . So consider $V_{U, y_2 \in \rho_2(x)}$ a base open of $E(\rho_2)$. By of $f$
its pre-image under $Rec(f)$ is
and hence a union of base opens.
()
Let $X$ be a and (def. ). Then the operations on
between the groupoid $Core(Cov(X))$ (example , def. ) whose are covering spaces over $X$, and whose are between these (def. ) and the groupoid $Rep(\Pi_1(X), Set)$ of (def. ) of the $\Pi_1(X)$ of $X$ (example ).
First we demonstrate a homotopy () of the form
To this end, given $\rho \in Rep(\Pi_1(X), Set)$ a , we need to exhibit in turn a homotopy () of permutation representations.
First consider what the right hand side is like: By def. of $Rec$ and def. of $Fib$ we have for every $x \in X$ an actual equality
To similarly understand the value of $Fib(Rec(\rho))$ on morphisms $[\gamma] \in \Pi_1(X)$, let $\gamma \colon [0,1] \to X$ be a representing in $X$. As in the proof of the path lifting lemma we find a of paths $\{\gamma_i\}_{i \in \{1,n\}}$ such that
regarded as morphisms $[\gamma_i]$ in $\Pi_1(X)$ they to $[\gamma]$:
each $\gamma_i$ factors through an open subset $U_i \subset X$ over which $Rec(\rho)$ trivializes.
Hence by of $Fib(Rec(\rho))$ it is sufficient to understand its value on these paths $\gamma_i$. But on these we have again by direct unwinding of the definitions that
This means that if we take
to be the above identification, then this is a homotopy/ as required.
It remains to see that these morphism $\eta_\rho$ are themselves natural in $\rho$, hence that for each morphism $\phi \colon \rho \to \rho'$ the diagram
commutes as a diagram in $Rep(\Pi_1(X), Set)$. Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all $x \in X$ the corresponding component diagram commutes. But by the above this is
and hence this means that the top and bottom horizontal morphism are in fact equal. Directz unwiinding of the definitions shows that this is indeed the case.
Now we demonstrate a homotopy () of the form
For $E \in Cov(X)$ a covering space, we need to exhibit a natural isomorphism of covering spaces of the form
Again by def. of $Rec$ and def. of $Fib$ the underlying set of $Rec(Fib(E))$ is actually equal to that of $E$, hence it is sufficient to check that this on underlying sets is a of .
By the assumption that $X$ is and , it is sufficient to check for $U\subset X$ an open path-connected subset and $x \in X$ a point with the property that $\pi_1(U,x) \to \pi_1(X,x)$ lands is constant on the trivial element, that the open subsets of $E$ of the form $U \times \{\hat x\} \subset p^{-1}(U)$ form a basis for the topology of $Rec(Fib(E))$. But this is the case by definition of $Rec$.
This proves the equivalence.
By example the covering space is connected and simply connected precisely if its representation is free and transitive. By the (prop. ) every of the $\pi_1(X)$ arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of $\pi_1(X)$ exists: The action of any group $G$ on itself, by left multiplication, is free and transitive.
$\,$
We now use the theorems established above to compute the of topological spaces in some basic examples. In particular we prove the archetypical example saying that the (prop. below).
$\,$
()
The $\pi_1$ of the $S^1$ is the additive group of :
and the isomorphism is given by assigning .
By example , the $\widehat{S^1}$ of $S^1$ is the
Since the is (this example) and (this example) the applies and gives that the of $\mathbb{R}^1$ over $S^1$ equals the automorphism group of its :
Moreover, as a corollary of the we have that the representation of a is given by the of the $\pi_1(S)$ on itself (this prop.).
But the of any group regarded as an on itself by left multiplication is canonically isomorphic to that group itself (by this example), hence we have
Therefore to conclude the proof it is now sufficient to show that
To that end, consider a of the form
Let $s \in S^1$ be any point, and consider the restriction of $f$ to the fibers over the :
By the property we have (via this example) a
Therefore, up to homeomorphism, the restricted function is of the form
By the of the this means that $f$ is equivalently given by its two components
By the of the above , the first component is fixed to be $pr_1$. Moreover, by the fact that $Disc(\mathbb{Z})$ is a it follows that the second component is a (by this example). Therefore, since the with a is a (via this example)
and since the disjoint summands $(0,1)$ are (this example), it follows that the second component is a on each of these summands (by this example).
Finally, since every function out of a is continuous, it follows in conclusion that the restriction of $f$ to the fibers over $S^1 \setminus \{s\}$ is entirely encoded in an of the set of
by
Now let $s' \in S^1$ be another point, distinct from $s$. The same analysis as above applies now to the restriction of $f$ to $S^1 \setminus \{s'\}$ and yields a function
Since
is an of $\mathbb{R}^1$, it follows that $f$ is unqiuely fixed by its restrictions to these two subsets.
Now unwinding the definition of $p$ shows that the condition that the two restrictions coincide on the intersection $S^1 \setminus \{s,s'\}$ implies that there is $n \in \mathbb{Z}$ such that $\phi(k) = k+ n$ and $\phi'(k) = k+n$.
This shows that $Aut_{Cov(S^1)}(\mathbb{R}^1) \simeq \mathbb{Z}$.
( of of the circle are in the )
The construction assigns to an of covering spaces over the $S^1$ with consisting of $n$ elements of elements the $\Sigma(n)$:
To see this we may without restriction choose a basepoint $x \in S^1$ so that a monodromy representation is equivalently a groupoid morphism of the form (prop. )
Since $\mathbb{Z}$ is the on a single generator, such as morphism is uniquely determined by the image of $1 \in \mathbb{Z}$. This is taken to some isomorphism of the set $p^{-1}(x)$. If we choose any identification $\phi \colon p^{-1}(x) \overset{\simeq}{\to} \{1, \cdots, n\}$, then this defines an element $\sigma \in \Sigma(n)$ in the :
Now if
is an isomorphism of covering spaces, then by the (prop. ) this corresponds bijectively to a homomorphism of representations
which in turn is by definition a homotopy (natural isomorphism) between the monodromy functors $Fib_{E_i} \;\colon\; B \mathbb{Z} \to Core(Set)$.
The combination of the naturality square of this natural isomorphism with the above identification yields the following diagram
The commutativity of the total rectangle says that the permutations $\sigma_1$ and $\sigma_2$ are related by conjugation with the element $\phi_2 \circ f\vert_{\{x\}} \circ \phi_1^{-1}$.
(three-sheeted covers of the circle)
Consider the three-sheeted of the .
By example these are, up to isomorphism, given by the of the elements of the $\Sigma(3)$ on three elements. These in turn are labeled by the structure of the elements (this prop.).
For the symmetric group on three elements there are three such classes
The corresponding covering spaces of the circle are shown in the graphics.
graphics grabbed from Hatcher
(…)
$\,$
This concludes the introduction to basic homotopy theory.
For introduction to more general and abstract homotopy theory see at .
An incarnation of in is . For introduction to that see at .
$\,$
A textbook account is in
Lecture notes include
$\,$
$\,$
Last revised on September 15, 2017 at 07:47:55. See the history of this page for a list of all contributions to it.