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From the Hopf theorem, we know that the degree classifies maps between spheres up to homotopy. Are more refined invariants (or useful notions of equivalence) that could distinguish between homotopic maps (and hence of the same degree) known?
For example, the degree of the identity and antipodal maps on $S^n$ are the same for $n$ odd, would any known (integer or algebraic) invariant distinguish between them?

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    What do you mean that could distinguish between maps of the same degree? Distinguish in what sense? – 2017-02-04
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    If $I$ is this invariant, two maps $f_1,f_2:S^n \to S^n$ having the same degree may be such that $I(f_1) \neq I(f_2)$. I am not sure I fully understood your question. – 2017-02-04
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    That is part of the question: a meaningful notion of equivalence. Pointwise equality, or equality up to a set of measure zero are I think pretty clearly not meaningful in this context. – 2017-02-05
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    I am not quite sure where these comments come from. A lot of research is low dimensional topology is aimed at differentiating spaces that are homotopy equivalent--admittedly, there is a clear notion of equivalence there (homeo or diffeo). I am simply asking whether there is something similar for maps on the sphere, or something that could be applied to it. – 2017-02-05
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    Being homotopy equivalent/homeomorphic/diffeomorphic arr equivalence relations between spaces. If you try to do something similar with continuous maps, you may want to distinguish between continuous maps and smooth maps, and ask whether two homotopic maps are smoothly homotopic. This is always the case, ie, the smooth homotopy relation is the same as the continuous homotopy relation. – 2017-02-05
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    Yes, I thought about that, and also about putting some notion of equivalence on domain/codomain of the function (e.g $f_1$ is eq. to $f_2$ if if $f_1 \circ \psi = \phi\circ f_2$ for some family of $\psi, \phi$, but no luck so far. Hence my posting here. – 2017-02-05
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    I edited the question to avoid being told once more that a notion of equivalence is needed. – 2017-02-05
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    If anyone understands the question, could you please propose edits? I made it as clear as I could, but with the exception of user17786, only received non-helpful comments and folks just seem to tack on to them. I understand that an invariant yields an equivalence relation, and this is exactly what I am asking for. – 2017-02-05
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    I don't understand why this was closed. I thought it was a perfectly good (and very interesting, to boot) question... Wasn't it clear that a meaningful and finer than homotopy equivalence relation was part of what was being ask for?! – 2017-02-05
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    I think there is a germ of an interesting question here. You have stated that you'd like an invariant/equivalence which distinguishes between, for example, the identity and antipodal maps for any sphere. The "equality" relation would certainly do this, and I guess some would call it useful, though you have stated that it is "pretty clearly not meaningful in this context": why? What would characterise a "meaningful" equivalence in this context? Perhaps some effort should be expended to describe what sorts of maps should remain equivalent under the desired equivalence(s). – 2017-02-05
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    The set of invariants in this case is the set of functions -- as large as what I started with. An integer invariant would be best. I don't know what the equivalence should be. I can make the following parallel: I know about $H_1$ and ask about an invariant that can distinguish spaces are in the same "1st homology class'' (i.e. two spaces are equiv. if they have same $H_1$). The definition of $H_2$ would make me happy. Asking me ``but for what equivalence relation'' is for me asking me my own question. This is why I find it difficult to expand on it. – 2017-02-05
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    Frankly, I find all this debate unusefully dense. The originals question quite clearly wanted to know what ways people have use and use to tell apart maps which are homotopic. Meaningful means, unsurprisingly, meaningful in the context where those people were working to begin with. It is a very natural question,have people needed to distinguish maps which are homotopic, and how did they do it? – 2017-02-05
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    Obviously, anyone minimally knowledgeable will be able to tell that the Identity relation is not a useful.kne in this context. – 2017-02-05
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    The only commonly used notion I can think of is whether a homotopy equivalence is a simple homotopy equivalence. This is not a useful enhancement in the simply connected case, though. – 2017-02-05
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    People also use positive homotopies, after endowing the sphere with structure so that this makes sense —for example, a lorentzian metric and restricting to timelike homotopies, and that sort of thing. – 2017-02-05
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    In the smooth category, and if the maps have restricted singularities (for example, isolated singular points which are non-degenerate be in some sense) be can try to go from one to the other without introducing worse degeneracies. This is interesting even if the codomain is contractible and one is only looking at immersions. – 2017-02-05
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    I am sure people have looked at homotopies involving finite energy (fix a Riemannian metric on the codomain and impose growth conditions on some action integral) "Controlled" homotopies, I think they call this. – 2017-02-05

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