TITLE: Other homotopy invariants?
QUESTION [8 upvotes]: 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 ...

REPLY [16 votes]: 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.