List of invariants that distinguish homotopy equivalent non-homeomorphic spaces

Usual homotopy-type invariants of the configuration spaces associated to a given space may be able to distinguish between spaces which are homotopy equivalent but not homeomorphic. For instance, Riccardo Longoni and Paolo Salvatore [Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375–380; MR2114713] distinguish between the lens spaces $L(7,1)$ and $L(7,2)$ by considering the universal covers of the (ordered) configuration spaces of $2$ points on these lens spaces, and showing that a certain Massey product vanishes for $\widetilde{F_2(L(7,1))}$ but not for $\widetilde{F_2(L(7,2))}$.

Sticking to the case of manifolds, the Kirby-Siebenmann invariant of a topological manifold $X$ is an element of $H^4(X;Z_2)$ that vanishes if $X$ admits a PL structure. (Having a PL structure is a topological invariant.) The simplest example, in some sense, is the manifold $*CP^2$ (or the Chern manifold) constructed by Freedman. It is homotopy equivalent to $CP^2$, but not homeomorphic to $CP^2$, since its Kirby-Siebenmann invariant is non-trivial.

In higher dimensions, the rational Pontrjagin classes are topological invariants by a fundamental theorem of Novikov. These can be varied more or less arbitrarily within a homotopy type, subject to the condition that the polynomial in the Pontrjagin classes (the L-polynomial) that determines the signature is constant. This is a key result of high-dimensional surgery theory; see Browder's book Surgery on Simply Connected Manifolds.

Let

$$ X^{\square\setminus\Delta}\ :=\ \{(x\ y)\in X^2 : x\ne y\} $$

be the deleted square of X. When $\ X\ $ is a topological space, then the deleted square (with the subspace topology induced by the square) is a topological invariant which is not a homotopy invariant. Then several other derived operations are likewise topologically but not homotopically invariant even when they are a composition of the deleted square and of a homotopically invariant operation--for instance, $\ \pi_1(X^{\square\setminus\Delta})\ $ is topologically but not homotopically invariant. This worked well for manifolds (in particular in Hirsch's hands).

EXAMPLE   Let $\ I\ $ be a closed interval, and $\ T\ $ be any finite tree with more than two endpoints; thus, $\ I\ $ and $\ T\ $ are homotopically equivalent (to the 1-point space). However $\ T^{\square\setminus\Delta}\ $ is connected, while $\ I^{\square\setminus\Delta}\ $ is disconnected.

We see that the topologically invariant operation of the deleted square, as well as the fundamental group of the deleted square, can distinguish between homotopically invariant spaces (even in the simple cases).