The motivation I was given for algebraic topology is to assign some algebraic objects as invariants to topological spaces. This way we can show that two spaces are not homeomorphic if they are assigned a different invariant. However it seems to me like in algebraic topology the invariants are always up to homotopy equivalence or maybe even only weak homotopy equivalence. This seems strange to me. Is there a reason why other kinds of algebraic invariants that can distinguish homotopy equivalent but not homeomorphic spaces not more widely studied? What is it about weak homotopy equivalence that is so special that seems to be the main focus of all algebraic topology?
(Why) Does Algebraic Topology only deal with topological spaces up to homotopy
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When Poincaré first envisioned algebraic topology, he envisioned it as a study of smooth manifolds under the equivalence relation if diffeomorphism [Analysis situs, pages 196-198]. Lefshetz [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem.
Algebraic topology developed in the PL category (Combinatorial Topology), because it was believed that this would give a useful avenue of attack on the differentiable case. Great algebraic topologists of the early 20th century (Reidemeister, Seifert, Schubert$\thinspace\ldots$) all worked with triangulated PL manifolds, and wrote good, precise, rigourous papers which are still valuable today (I think that Schubert's Topologie is one of the greatest topology textbooks ever written). They were gluing together, and subdividing, finite collections of simplices; their group theory was combinatorial; and the subject as a whole had a highly combinatorial flavour to it. And it was great! Everything was explicit, and there was no need for fudgy handwavy `corners can be rounded' type arguments to be thrown around. In my opinion, simplicial complexes continue to be the best setting to work explicitly with linking forms, for example.
In the 1950's and 1960's, with work of Smale, Thom, Milnor, Hirsch, and others, honest smooth algebraic topology became possible, and the relationship between PL and smooth categories was clarified. And after that, people began switching back and forth at will when it was possible to do so, and, with the basic groundwork for algebraic topology established in both categories, the combinatorial flavour of the subject became dulled. Combinatorial Group Theory went off and became its own subject, and the majority of topologists no longer saw the need to mess about with explicit triangulations of manifolds- they just worked directly with invariants of the chain complex. And CW complexes became used instead of simplicial complexes, for example because the dual cell subdivision of a simplicial complex need no longer be a simplicial complex.
But "combinatorial topology" in its former sense still very much exists. An it's not going to go away. To programme topology into a computer for example, you need an explicit triangulation, and the work is all combinatorial and PL. See for example Matveev's Algorithmic topology and classification of 3-manifolds. The constructivist argument would be that `real world' manifolds (whatever that means) are PL.
Yes, there are such structures, e.g. a cohomology ring:
https://en.wikipedia.org/wiki/Cohomology_ring
Actually they are quite important. For example by analysing the cohomology ring of spheres it can be deduced that there is no topological group structure on spheres except for dimensions $0, 1, 3$ and somewhat weaker (i.e. not associative) topological group structure in dimension $7$.
Actually in case of topological groups this goes even further. The cohomology ring (of a topological group) becomes a Hopf algebra which is a very rich algebraic structure: it's a vector space with multiplication and comultiplication. So it has at least four operators ($+$, scalar multiplication, vector multiplication, comultiplication).
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Algebraic topology would like to classify spaces up to homeomorphism (as well as some other categories of equivalence, like piecewise-linear homeomorphism) rather than just up to homotopy equivalence, but the former turns out to be much harder. The reason for the focus on the latter is just that homology and cohomology are powerful invariants that are generally reasonable to compute, and their invariance under homotopy equivalence is often good enough to prove that two spaces aren't homeomorphic. They pop up in group theory and in algebraic and differential geometry, in results like the Lefschetz fixed-point theorem and the Whitehead theorem, and a lot of other places.
The analogy I have in the back of my head is to equivalence of central extensions of groups: We actually want to classify them up to isomorphism, but that turns out to be much harder than classifying them up to equivalence via group cohomology, and the latter is often good enough for particular results. Or just consider group theory itself: We'd like to classify groups completely up to isomorphism, but that's hard to do, and there are a lot of other interesting questions along the way.
But as useful as homology and cohomology are, they're not the full extent of algebraic topology. The classification of $3$-dimensional lens spaces $L(p, q)$ up to homotopy equivalence is straightforward enough, but determining them up to homeomorphism is more complicated. (In particular, they give examples of spaces that are homotopy equivalent but not homeomorphic, beyond trivial ones like contractible spaces.) Historically, this was done with more complicated algebraic invariants like Reidemeister torsion, and that led to a variety of results in obstruction theory, higher cohomology operations, and surgery theory. I could spend pages talking about more algebraic topology constructions beyond (simplicial, cellular, etc.) homology and cohomology, so I'll limit myself to mentioning the cobordism ring, Donaldson's theorem on $4$-manifolds, and Milnor's construction of an exotic $S^7$ by the Pontryagin class.
These sorts of invariants are squarely part of algebraic topology, but they're not usually covered in introductory courses like Hatcher due to the algebraic and topology prerequisites involved; besides, at that point, you're firmly in algebraic topology (or geometric topology, but at least topology) territory, and there are fewer broadly applicable topics like general cohomology.