Theory

Homotopy type and homology

Homology can be used to distinguish topological spaces or to measure the topological complexity of a space. From the perspective of computational complexity theory, homology is very easy to compute.

Theorem: Homology depends functionally on $|\mathcal{K}|$

The isomorphism types of the homology groups $H_i(\mathcal{K};\mathbb{Z})$ of a simplicial complex $\mathcal{K}$ depend only on $|\mathcal{K}|$, the underlying topological space.

More precisely, a homeomorphism of topological spaces induces an isomorphism of homology groups.

This theorem implies that any two homeomorphic spaces have isomorphic homology groups.

• %& For a topological space $X$, we write $H_i(X;\mathbb{Z})$ for the homology groups of $X$ computed using any simplicial complex structure provided on $X$.
  • If $X$ does not have a simplicial complex structure, or $\Delta$-complex structure, one can still define singular homology for $X$, and this homology still depends functionally only on the homeomorphism type.
  • Singular homology is isomorphic to simplicial and $\Delta$-homology when the latter can be defined.

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There is an equivalence relation between spaces that is much ‘coarser’ than homeomorphism, namely homotopy equivalence.

Homotopy equivalence

A homotopy equivalence between topological spaces $X$ and $Y$ is a pair $(f,g)$ of continuous maps:

such that $f\circ g$ is homotopic to ${\mathrm{Id}}_X$ and $g\circ f$ is homotopic to ${\mathrm{Id}}_Y$.

We write $X\simeq Y$ when $X$ and $Y$ are homotopy equivalent.

A homotopy from $f\circ g$ to ${\mathrm{Id}}_X$ would be a continuous map $F:X\times [0,1]\to X$ such that $F(x,0)=(f\circ g)(x)$ and $F(x,1)={\mathrm{Id}}_X(x)=x$. Analogously for a homotopy $g\circ f$ to ${\mathrm{Id}}_Y$.

A homeomorphism is actually the special case of a homotopy equivalence where the homotopies $f\circ g\stackrel{\sim }{⟶}{\mathrm{Id}}_X$ and $g\circ f\stackrel{\sim }{⟶}{\mathrm{Id}}_Y$ are both the identity maps.

Theorem: Homology depends functionally on homotopy type

The isomorphism types of the homology groups $H_i(\mathcal{K};\mathbb{Z})$ of a topological space $|\mathcal{K}|$ depend only on the homotopy type of the space.

More precisely, a homotopy equivalence of topological spaces induces an isomorphism of homology groups.

We will see a partial justification of this fact later, after discussing simplicial maps and chain maps.

Contractible spaces

A topological space is contractible if it is homotopy equivalent to a point.

Illustration - convex sets and simplices are contractible

The same homotopy formula, namely $F(x,t)=tx$, can be used to show that any ball in ${\mathbb{R}}^n$ is contractible, and indeed any convex set in ${\mathbb{R}}^n$ is contractible.

• ! Therefore, any simplex is contractible.