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In algebraic topology one investigates topological spaces $X$ by looking at

  • their singular homology groups $H_n(X)$,
  • their singular cohomology groups $H^n(X)$, and
  • their homotopy groups $\pi_n(X)$.

However, I never saw any other invariant than (co)homology or homotopy groups.

Question: Is it possible that someone comes up with a completely new invariant? Or is there a theorem which states something in the direction that the above invariants are sufficient (for some purposes)?

user475784
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    An interesting result is Brown's representability theorem. https://en.wikipedia.org/wiki/Brown%27s_representability_theorem – Randall Jan 25 '22 at 17:06
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    Aside from the groups themselves, there are other structures that one can use to distinguish spaces. For example, the direct sum of the cohomology groups have a ring structure, and the higher homotopy groups have the structure of $\pi_1$-modules. In addition, cohomology with $\mathbb{Z}_2$ coefficients has the structure of a module over the Steenrod algebra - see this question for an example of two spaces which cannot be distinguished using the prior structures, but can by using this additional one. – Michael Albanese Jan 25 '22 at 17:12
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    There are also extraordinary cohomology theories like K-theory or cobordism theories. – ThePuix Jan 25 '22 at 17:48
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    Related: https://mathoverflow.net/questions/335112/other-homotopy-invariants. – Qi Zhu Jan 25 '22 at 18:35

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