To add a little to Ryan's answer:

This topic is maybe not exactly a part of algebraic topology. It's more like an area of application of algebraic to certain important special classes of spaces.

When $A$ is a subcomplex of $B$ (both of them finite) and $B$ collapses to $A$, then the inclusion map $A\to B$ is said to be a simple homotopy equivalence, as is any left inverse of such an inclusion, and more generally any map between finite complexes that is homotopic to a composition of such things. A homotopy equivalence $A\to B$ between finite complexes determines an element of the Whitehead group $Wh(G)$ of the fundamental group $G=\pi_1(A)$ (a certain abelian group that depends functorially on $G$ -- the quick definition is take the direct limit of $GL_n(Z[G])$ as $n$ goes to infinity, abelianize, and kill the the invertible $1\times 1$ matrices $g\in G$ and $-1$). It is simple if and only if this element is zero. (Sometimes the latter is taken as definition of simple.) The group $Wh(1)$ is trivial, so every homotopy equivalence between simply-connected finite complexes is simple. The house with two rooms shows that the inclusion of a subcomplex can be simple even if there is no collapse; there is a larger complex collapsing both to the house and to the point.

The question of whether a homotopy equivalence is simple is unchanged by subdivision of a complex, so sometimes you can prove that a given homotopy equivalence is not homotopic to any simplicial or cellular isomorphism by proving that it has nontrivial Whitehead torsion. If you can enumerate all the homotopy equivalences from $A$ to $B$ and find that all of them have nontrivial torsion, then the spaces are really different. Eventually the toplogical invariance of Whitehead torsion was proved, making for stronger statements.

$Wh(G)$ is trivial for lots of (potentially for all) torsion-free groups $G$, but usually nontrivial for finite $G$.

To add a little to Ryan's answer:

This topic is maybe not exactly a part of algebraic topology. It's more like an area of application of algebraic topology to certain important special classes of spaces.

When $A$ is a subcomplex of $B$ (both of them finite) and $B$ collapses to $A$, then the inclusion map $A\to B$ is said to be a simple homotopy equivalence, as is any left inverse of such an inclusion, and more generally any map between finite complexes that is homotopic to a composition of such things. A homotopy equivalence $A\to B$ between finite complexes determines an element of the Whitehead group $Wh(G)$ of the fundamental group $G=\pi_1(A)$ (a certain abelian group that depends functorially on $G$ -- the quick definition is take the direct limit of $GL_n(Z[G])$ as $n$ goes to infinity, abelianize, and kill the the invertible $1\times 1$ matrices $g\in G$ and $-1$). It is simple if and only if this element is zero. (Sometimes the latter is taken as definition of simple.) The group $Wh(1)$ is trivial, so every homotopy equivalence between simply-connected finite complexes is simple. The house with two rooms shows that the inclusion of a subcomplex can be simple even if there is no collapse; there is a larger complex collapsing both to the house and to the point.

The question of whether a homotopy equivalence is simple is unchanged by subdivision of a complex, so sometimes you can prove that a given homotopy equivalence is not homotopic to any simplicial or cellular isomorphism by proving that it has nontrivial Whitehead torsion. If you can enumerate all the (homotopy classes of) homotopy equivalences from $A$ to $B$ and you find that all of them have nontrivial torsion, then the spaces are really different. Eventually the topological invariance of Whitehead torsion was proved, making for stronger statements.

$Wh(G)$ is trivial for lots of (potentially for all) torsion-free groups $G$, but is usually nontrivial for finite $G$.