Contents

# Contents

## Idea

A pointed topological space (often pointed space, for short) is a topological space equipped with a choice of one of its points (elements).

Although this concept may seem simple, pointed topological spaces play a central role for instance in algebraic topology as domains for reduced generalized (Eilenberg-Steenrod) cohomology theories and as an ingredient for the definition of spectra.

One reason why pointed topological spaces are important is that the category which they form is an intermediate stage in the stabilization of homotopy theory (the classical homotopy theory of topological spaces) to stable homotopy theory:

The category of pointed topological spaces has a zero object (the point space itself) and the canonical tensor product on pointed spaces is the smash product, which is non-cartesian monoidal category, in contrast to the plain product of topological space.

## Definition

A pointed topological space is a topological space $(X,\tau)$ equipped with a choice of point $x \in X$. A homomorphism between pointed topological space $(X,x)$ $(Y,y)$ is a continuous function $f \colon X \to Y$ which preserves the chosen basepoits in that $f(x) = y$.

### The category of pointed topological spaces

Stated in the language of category theory, this means that pointed topological spaces are the pointed objects in the category Top of topological spaces. This is the coslice category $Top^{\ast/}$ of topological spaces “under” the point space $\ast$:

an object in $Top^{\ast/}$ is equivalently a continuous function $x \colon \ast \to (X,\tau)$, which is equivalently just a choice of point in $X$, and a morphism in $Top^{\ast/}$ is a morphism $f \colon X \to Y$ in Top (hence a continuous function), such that this triangle diagram commutes

$\array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} && \\ X && \underset{f}{\longrightarrow} && Y }$

which equivalently means that $f(x) = y$.

###### Definition

The forgetful functor $Top^{\ast/} \to Top$ has a left adjoint given by forming the disjoint union space (coproduct in Top) with a point space (“adjoining a base point”), this is denoted by

$(-)_+ \coloneqq (-) \sqcup \ast \;\colon \; Top \longrightarrow Top^{\ast/} \,.$

### Wedge sum and Smash product

###### Example

Given two pointed topoligical spaces $(X,x)$ and $(Y,y)$, then:

1. their Cartesian product in $Top^{\ast/}$ is simply their product topological space $X \times Y$ equipped with the pair of basepoints $(X\times Y, (x,y))$;

2. their coproduct in $Top^{\ast/}$ has to be computed using the second clause in this prop.: since the point $\ast$ has to be adjoined to the diagram, it is given not by the coproduct in $Top$ (which is the disjoint union space), but by the pushout in $Top$ of the form:

$\array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,.$

This is called the wedge sum operation on pointed objects.

This is the quotient topological space of the disjoint union space under the equivalence relation which identifies the two basepoints:

$X \vee Y \;\simeq\; (X \sqcup Y)/(x \sim y)$

Generally for a set $\{(X_i,x_i)\}_{i \in I}$ of pointed topological spaces

1. their product is formed in Top, as the product topological space with the Tychonoff topology, with the tuple $(x_i)_{i \in I} \in \underset{i \in I}{\prod} X_i$ of basepoints being the new basepoint;

2. their coproduct is formed by the colimit in $Top$ over the diagram with a basepoint adjoined, and is called the wedge sum $\vee_{i \in I} X_i$, which is the quotient topological space of the disjoint union space with all the basepoints identified:

$\underset{i \in I}{\vee} X_i \;\simeq\; \left(\underset{i \in I}{\sqcup} X_i\right)/(x_i \sim x_j)_{i,j \in I} \,.$
###### Example

For $X$ a CW-complex, then for every $n \in \mathbb{N}$ the quotient of its $n$-skeleton by its $(n-1)$-skeleton is the wedge sum, def. , of $n$-spheres, one for each $n$-cell of $X$:

$X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,.$
###### Definition

The smash product of pointed topological spaces is the functor

$(-)\wedge(-) \;\colon\; Top^{\ast/} \times Top^{\ast/} \longrightarrow Top^{\ast/}$

given by

$X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,,$

hence by the pushout in $Top$ of he frm

$\array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,.$

In terms of the wedge sum from def. , this may be written concisely as the quotient space (this def) of the product topological space by the subspace constituted by the wedge sum

$X \wedge Y \simeq \frac{X\times Y}{X \vee Y} \,.$

t $\,$

symbolnamecategory theory
$X \times Y$product spaceproduct in $Top^{\ast/}$
$X \vee Y$wedge sumcoproduct in $Top^{\ast/}$
$X \wedge Y = \frac{X \times Y}{X \vee Y}$smash producttensor product in $Top^{\ast/}$
###### Example

For $X, Y \in Top$, with $X_+,Y_+ \in Top^{\ast/}$, def. , then

• $X_+ \vee Y_+ \simeq (X \sqcup Y)_+$;

• $X_+ \wedge Y_+ \simeq (X \times Y)_+$.

###### Proof

By example , $X_+ \vee Y_+$ is given by the colimit in $Top$ over the diagram

$\array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,.$

This is clearly $A \sqcup \ast \sqcup B$. Then, by definition

\begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned}
###### Example

Let $I \coloneqq [0,1] \subset \mathbb{R}$ be the closed interval with its Euclidean metric topology.

Hence

$I_+ \in Top^{\ast/}$

is the interval with a disjoint basepoint adjoined, def. .

Now for $X$ any pointed topological space, then the smash product (def. )

$X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)$

is the reduced cylinder over $X$: the result of forming the ordinary cylinder over $X$, and then identifying the interval over the basepoint of $X$ with the point.

(Generally, any construction in $Top$ properly adapted to pointed spaces is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

Just like the ordinary cylinder $X\times I$ receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top$, so the reduced cyclinder receives a canonical injection from the coproduct $X \sqcup X$ formed in $Top^{\ast/}$, which is the wedge sum from example :

$X \vee X \longrightarrow X \wedge (I_+) \,.$

### Mapping (co-)cones

Recall that the cone on a topological space $X$ is the quotient space of the product space with the closed interval

$Cone(X) = (X \times [0,1])/( X \times \{0\} ) \,.$

If $X$ is pointed with basepoint $x \in X$, then the reduced cone is the further quotient by the copy of the interval over the basepoint

$Cone(X,x) = Cone(X) / ( \{x\} \times [0,1] ) \,.$

For $f \colon X \to Y$ a continuous function, then

1. the mapping cylinder of $f$ is the attachment space

$Cyl(f) \coloneqq Y \cup_f Cyl(X)$
2. the mapping cone of $f$ is the attachment space

$Cone(f) \coloneqq Y \cup_f Cone(X)$

accordingly if $f \colon X \to Y$ is a continuous function between pointed spaces which preserves the basepoint, then the analogous construction with the reduced cylinder and the reduce cone, respectively, yield the reduced mapping cyclinder and the reduced mapping cone.

We now say this again in terms of pushouts:

###### Definition

For $f \colon X \longrightarrow Y$ a continuous function between pointed spces, its reduced mapping cone is the space

$Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y$

in the colimiting diagram

$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,,$

where $Cyl(X)$ is the reduced cylinder from def. .

###### Proposition

The colimit appearing in the definition of the reduced mapping cone in def. is equivalent to three consecutive pushouts:

$\array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,.$

The two intermediate objects appearing here are called

• the plain reduced cone $Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X)$;

• the reduced mapping cylinder $Cyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y$.

###### Definition

Let $X \in Top^{\ast/}$ be any pointed topological space.

The mapping cone, def. , of $X \to \ast$ is called the reduced suspension of $X$, denoted

$\Sigma X = Cone(X\to\ast)\,.$

Via prop. this is equivalently the coproduct of two copies of the cone on $X$ over their base:

$\array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,.$

This is also equivalently the cofiberf of $(i_0,i_1)$, hence (example ) of the wedge sum inclusion:

$X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,.$
###### Proposition

The reduced suspension objects (def. ) induced from the standard reduced cylinder $(-)\wedge (I_+)$ of example are isomorphic to the smash product (def. ) with the circle] (the [[1-sphere?)

$cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,,$
###### Proposition

For $f \colon X \longrightarrow Y$ a morphism in Top, then its unreduced mapping cone with respect to the standard cylinder object $X \times I$ def. , is isomorphic to the reduced mapping cone, of the morphism $f_+ \colon X_+ \to Y_+$ (with a basepoint adjoined) with respect to the standard reduced cylinder:

$Cone'(f) \simeq Cone(f_+) \,.$
###### Proof

By example , $Cone(f_+)$ is given by the colimit in $Top$ over the following diagram:

$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,.$

We may factor the vertical maps to give

$\array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,.$

This way the top part of the diagram (using the pasting law to compute the colimit in two stages) is manifestly a cocone under the result of applying $(-)_+$ to the diagram for the unreduced cone. Since $(-)_+$ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit $Cone'(f)_+$ as shown. The remaining pushout then contracts the remaining copy of the point away.

## Properties

Most of the relevant constructions on pointed topological spaces are immediate specializations of the general construction discussed at pointed object.