Other homotopy invariants?

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 ...

• The problem with using spaces other than suspensions is that (based or unbased) homotopy classes of maps out of them do not typically form a group. The only $n$-torus which is a suspension is $n=1$ case, i.e., the circle. – John Klein Jun 30 '19 at 4:30
• A similar question was asked here mathoverflow.net/q/37792/8103 – Mark Grant Jun 30 '19 at 6:12
• The idea of using strict higher groupoids as invariants of some structured spaces is discussed in my paper "Modelling and computing homotopy types : I" available from www,groupoids.org.uk/gpdsweb.html. – Ronnie Brown Jun 30 '19 at 9:24
• I suppose it depends on what you mean by "simple space". Singular bordism and singular homology would likely fit in most definitions of "simple space(s)". – Ryan Budney Jul 1 '19 at 1:01

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.
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).