Contents

group theory

# Contents

## Idea

The fundamental group $\pi_1(X,x)$ of a pointed topological space $(X,x)$ is the group of based homotopy classes of loops at $x$, with multiplication defined by concatenation (following one path by another).

This is also called the first homotopy group of $X$.

The notion of fundamental group $\pi_1(X,x)$ generalises in one direction to the fundamental groupoid $\Pi_1(X)$, or in another direction to the homotopy groups $\pi_n(X,a)$ for $n \in \mathbb{N}$. Both of this is contained within the fundamental ∞-groupoid $\Pi(X)$.

## Definition

###### Definition

For $X$ a topological space and $x : * \to X$ a point. A loop in $X$ based at $x$ is a continuous function

$\gamma : \Delta^1 \to X$

from the topological 1-simplex, such that $\gamma(0) = \gamma(1) = x$.

A based homotopy between two loops is a homotopy

$\array{ \Delta^1 \\ \downarrow^{\mathrlap{(id,\delta_0)}} & \searrow^{\mathrlap{f}} \\ \Delta^1 \times \Delta^1 &\stackrel{\eta}{\to}& X \\ \uparrow^{\mathrlap{(id,\delta_1)}} & \nearrow_{\mathrlap{g}} \\ \Delta^1 }$

such that $\eta(0,-) = \eta(1,-) = x$.

###### Proposition

This notion of based homotopy is an equivalence relation.

###### Proof

This is directly checked. It is also a special case of the general discussion at homotopy.

###### Definition

Given two loops $\gamma_1, \gamma_2 : \Delta^1 \to X$, define their concatenation to be the loop

$\gamma_2 \cdot \gamma_1 : t \mapsto \left\{ \array{ \gamma_1(2 t) & ( 0 \leq t \leq 1/2 ) \\ \gamma_2(2 (t-1/2)) & (1/2 \leq t \leq 1) } \right. \,.$
###### Proposition

Concatenation of loops respects based homotopy classes where it becomes an associative, unital binary pairing with inverses, hence the product in a group.

###### Definition

For $X$ a topological space and $x \in X$ a point, the set of based homotopy equivalence classes of based loops in $X$ equipped with the group structure from prop. is the fundamental group or first homotopy group of $(X,x)$, denoted

$\pi_1(X,x) \in Grp \,.$

Hence if we write $[\gamma] \in p_1(X,x)$ for the based homotopy class of a loop $p$, then then group operation is

$[\gamma_1] \cdot [\gamma_2] \coloneqq [\gamma_1 \cdot \gamma_2] \,.$
###### Definition

A topological space whose fundamental group is trivial is called a simply connected topological space.

###### Definition

Conversely, a topological space whose only non-trivial homotopy group is the fundamental group is called an Eilenberg-MacLane space denoted $K(\pi_1(X), 1)$.

## Properties

### Naturality

###### Proposition

If a topological space $X$ is path-connected space, then all of the fundamental groups $\pi_1(X,x)$ are isomorphic, for all choices of base points $x \in X$.

###### Proof

For $x_0, x_1 \in X$ any two basepoints, there is by assumption a path connecting them, hence a continuous function

$p : \Delta^1 \to X$

such that $p(0) = x_0$ and $p(1) = x_1$.

Write $\bar p \coloneqq (p(1-(-)))$ for the same path with the orientation reversed. Then for $\gamma_0$ any loop based at $x_0$, the concatenation $[ p \cdot (\gamma_0 \cdot \bar p) ]$, def. yield a loop based at $x_1$ (obtained from $[\gamma_0]$ by conjugation with $[p]$).

It is immediate to check that this induces an isomorphism

$Ad_{[p]} : \pi_1(X,x_0) \to \pi_1(X,x_1) \,.$
###### Remark

Therefore one sometimes loosely speaks of ‘the’ fundamental group of a connected space. But beware that the isomorphism in the above construction is not unique. Therefore forming fundamental groups is not a functor on connected spaces.

###### Proposition

It is, however, a functor on pointed topological spaces: $\pi_1(-) :$ Top ${}^{*/} \to$ Grp.

### Relation to singular homology

The fundamental group is in general non-abelian (i.e. is not an abelian group), the Examples below. For a connected topological space $X$, its abelianization is equivalent to the first singular homology group

$\pi_1(X,x)^{ab} \simeq H_1(X) \,.$

See at singular homology – Relation to homotopy groups for more on this.

### Relation to universal covers and Galois groups

There is a relation to universal covers: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space. This is the topic of the étale fundamental group, also referred to at Chevalley fundamental group, see there for more.

In particular in algebraic geometry and arithmetic geometry this essentially identifies the concept of fundamental group with that of Galois groups. For this reason one also speaks of the algebraic fundamental group in this context. See at Galois theory for more on this.

In Grothendieck's Galois theory, the role of the basepoint is replaced by considering a ‘fibre functor’ $F:\mathcal{C}\to Sets$ or to $FinSets$, where $\mathcal{C}$ is the category of coverings of the given space. This theory extends to other situations and the term algebraic fundamental group is used in particular for the case of schemes (of a suitable type); see (SGA1).

## Generalizations

### Non-locally ‘nice’ spaces and ‘generalised’ spaces

The definition of fundamental group in terms of homotopy classes of loops at a base point does not work well for the spaces that occur in algebraic geometry, nor for many spaces considered in analysis as there may be very few loops. For instance, for a scheme there are in general very few paths, and Grothendieck gave a definition of a fundamental group in SGA1 which is closely related to the Galois groups of number theory, but in cases where both the path-based group and this algebraic fundamental group make sense, the algebraic form tends to be related to the profinite completion of the topological fundamental group; see the example in that entry.

A similar type of construction gives the fundamental group of a topos. Other related forms include a Čech version of the fundamental group used in shape theory, and linked to Čech homology groups of a compact space.

The notion of fundamental group generalizes to that of fundamental groupoid in both the loop based theory and in Grothendieck's Galois theory as described in SGA1. In this form it has been used to give generalisations for simplicial profinite spaces in work by Quick and to pro-spaces in work of Isaksen.

### Proper fundamental groups

In the context of proper homotopy theory there are two related fundamental groups for single ended spaces.

## Examples

###### Example

(Euclidean space is simply connected)

For $n \in \mathbb{N}$, let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space with its metric topology. Then for every point $x \in \mathbb{R}^n$ the fundamental group is trivial:

$\pi_1(\mathbb{R}^n, x) = 1 \,.$
###### Proof

Let

$\gamma \;\colon\; [0,1] \longrightarrow \mathbb{R}^n$

be loop at $x$, hence a continuous function with $\gamma(0) = x$ and $\gamma(1) = x$. Using the real vector space structure on $\mathbb{R}^n$, we may define the function

$\array{ [0,1] \times [0,1] &\overset{\eta}{\longrightarrow}& \mathbb{R}^n \\ (t,s) &\mapsto& x + s (\gamma(t) - x) } \,.$

This is a continuous function, since it is the composite of the continuous function $(id_{[0,1]} \times \gamma) \;\colon\; (t,s) \mapsto (\gamma(t),s)$ (which is continuous as the product of two continuous functions) and the function $(v,s) \mapsto x + s ( v - x )$ (which is continuous since polynomials are continuous).

Moreover, by construction we have

$\eta(-,1) = \gamma(-) \phantom{AAAA} \eta(-,0) = const_x \,.$

Therefore this is a homotopy from $\gamma$ to the constant loop at $x$.

###### Example

By definition , the fundamental group of every simply connected topological space is trivial.

###### Example
$\pi_1(S^1) \simeq \mathbb{Z} \,.$
###### Remark

An instructive formalization of this basic statement in homotopy type theory is in (Shulman).

###### Example

By definition , the fundamental group of any Eilenberg-MacLane space $K(G,1)$ is $G$: $\pi_1(K(G,1)) = G$.

• Jesper Møller, The fundamental group and covering spaces (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 2.5 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

Discussion from the point of view of Galois theory is in

Isaksen’s work is

• D. C. Isaksen, A model structure on the category of pro-simplicial sets,

Trans. Amer. Math. Soc., 353, (2001), 2805–2841

whilst Quick’s is in

• G. Quick, Profinite homotopy theory, Documenta Mathematica, 13, (2008), 585–612.

Proof that the fundamental group of the circle is the integers in homotopy type theory is in

the HoTT-Coq-code is at