Other homotopy invariants?

Question

The idea of using maps from a sequence of simple standard objects into a topological space $X$ as $probes$ to explore its topology is ubiquitous. One organizes these maps into equivalence classes in such a way that the collection of classes acquires a nice algebraic structure. These algebraic invariants then serve to recognize $X$ or distinguish it from others.

One such sequence is, of course, pointed $n$-spheres, homotopy classes of maps from which yield homotopy groups, $\pi_n (X)$.

Has it been useful to consider other sequences of simple spaces for construction of invariants, e.g., homotopy classes of maps from $n$-tori, or from genus $n$ tori? Or can these always be simply expressed in terms of homotopy groups, and are, therefore, redundant? Or too hard to compute? Or lack good properties? Or ...

Answer 1

A hot topic for 20 years starting in the mid 1980's was the exploration of spaces by `probing' them with the spaces $BV$ where $V$ is a group of the form $(\mathbb Z/p)^n$ with $p$ a prime. It is a theorem of Jean Lannes, building on work of Haynes Miller, that, under remarkably mild hypotheses, the set of maps $[BV,X]$ is a very computable functor of $H^*(X;\mathbb Z/p)$, viewed as an algebra equipped with Steenrod operations. Even better: $H^*(Map(BV,X);\mathbb Z/p) = T_VH^*(X;\mathbb Z/p)$, where $T_V$ is a wonderful algebraic functor discovered by Lannes.

This had many applications to a wide range of problems, ranging from the classification of polynomial rings that can be realized as the cohomology of a space, to the theorem that, if $H^*(X;\mathbb Z/p)$ has infinite total dimension as a $\mathbb Z/p$ vector space, then it must also be infinitely generated as a module over the Steenrod algebra.

Answer 2

Dually the story is also very beautiful. Instead of spaces mapping into a reasonable space $X$, one can look at mappings of $X$ into spaces. Of course for any space $Y$ the homotopy classes $ X\rightarrow Y$ is a homotopy invariant for $X$. But this is very hard to compute in general, because the set $[X,Y]$ does not have any structure. However, if the space(s) $Y$ have structure (e.g. they sit in a spectrum) much more structure is available, which allows one to compute these homotopy classes. This is the case for any (generalized) cohomology theory: This includes singular cohomology (maps into Eilenberg-Maclane spaces), $K$-theory (maps into Fredholm operators), cohomotopy (maps into spheres), bordism (maps into universal thom spaces) and much more. To obtain generalized homological invariants one can look at homotopy groups of $Y\wedge X$, which is not exactly what you asked.

There are also some general statements what $Y$ should be such that $[X,Y]$ and $[Y,X]$ form the structure of a group. This is what John Klein is alluding to above. I don't know too many invariants that are used daily which do not arise in this manner (except maybe the Lyusternik-Schnirelmann category).