Homotopy equivalent topological spaces

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Proper stub, could use some more references regarding terminology

Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces, we say $X$ is homotopy equivalent to $Y$, or $X$ and $Y$ have the same homotopy type, written $X\simeq Y$, if:

• $\exists f\in$$C(X,Y)$$\exists g\in C(Y,X)\big[(g\circ f\simeq$$\text{Id}_X$$)\wedge(g\circ f\simeq \text{Id}_Y)\big]$
  • Here $C(X,Y)$ denotes the set of continuous maps from $X$ and $f\simeq g$ denotes the relation of homotopy of maps - that is in this case freely homotopic
  • In words, there exist two continuous maps, $f:X\rightarrow Y$ and $g:Y\rightarrow X$ such that $(g\circ f):X\rightarrow X$ is freely homotopic to $\text{Id}_X:X\rightarrow X$ (the identity map on $X$) with $\text{Id}_X:x\mapsto x$ and $(f\circ g):Y\rightarrow Y$ is again freely homotopic to $\text{Id}_Y:Y\rightarrow Y$ by $\text{Id}_Y:y\mapsto y$

Terminology

Let $f:X\rightarrow Y$ be a continuous map (so $f\in C(X,Y)$ in other words) and let $g:Y\rightarrow X$ be another continuous map (so $g\in C(Y,X)$, as before), then:

• if $(g\circ f)\simeq \text{Id}_X$ and $(f\circ g)\simeq\text{Id}_Y$ (so $X\simeq Y$, as is the topic of this page) then we may say:
  • $g$ is a homotopy inverse for $f$
  • $f$ is a homotopy equivalence (as it has a homotopy inverse, namely $g$)
    • Note also that if $X\simeq Y$ with $f$ and $g$ then $Y\simeq X$ with $g$ and $f$ as the maps, leading to:
      • $f$ is a homotopy inverse for $g$ and
      • $g$ is a homotopy equivalence (as it has a homotopy inverse, namely $f$)

    We will later see that homotopy equivalence of topological spaces is an equivalence relation, so if $X\simeq Y$ then $Y\simeq X$, as we've basically just shown, the symmetric property of an equivalence relation is easy to see!

See next

• Homotopy equivalence of topological spaces is an equivalence relation - the relation of $X\simeq Y$ is an equivalence relation
• A deformation retract induces a homotopy equivalence
  • If $A$ (as a topological subspace of $(X,\mathcal{ J })$) is a deformation retract of $X$ then $X\simeq A$
  • TODO: A strong deformation retraction must be a homotopy equivalence too, as it itself is an instance of a a weak deformation retraction - add this somewhere Alec (talk) 21:13, 24 April 2017 (UTC)
• Homotopy invariance of the fundamental group - homotopically equivalent spaces have isomorphic fundamental groups
  • If $f:X\rightarrow Y$ is a homotopy equivalence (so $X\simeq Y$ through $f$ and some other map which would be its homotopy inverse) then:
    • $\forall p\in X[\pi_1(X,p)\cong_{f_*}\pi_1(Y,f(p))]$ where $\cong$ denotes isomorphism of groups and $f_*:\pi_1(X,p)\rightarrow\pi_1(Y,f(p))$ the fundamental group homomorphism induced by a continuous map