# On the functor H

Question: What can you deduce about $f$ by examining $H_{\ast}f?$

Detailed version of the question:

Let $H$ be a homology theory which satisfy the Eilenberg-Steenrod-Milnor axioms (see, for instance, Bredon's topology page 183). Since, in particular, it is a functor every 'continuous' map $f\colon X\rightarrow Y$ induces a homomorphism $H_{\ast}f$ on the corresponding groups. (again see Bredon for the construction of this homomorphism).

Now, my question is the following: Suppose we are using metic spaces, and , f is Lipschitz, or biLipschitz, or Hölder continuous, or, say $f$ is a positive kernel on $X.$ There are many different properties that $f$ can satisfy.You are welcome to add your favorite property of continuous maps which are homotopy invariant or not

Is it possible to extract these information about $f$ by examining the group homomorphisms $H_{\ast}f?$ How?So, I am interested in passing from group homomorphisms to functions Are there different homology theories that one can construct so that these abstract homomorphisms give information about $f$?

Thank you.
Edit1: I ve added the bold text

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Homology theories are homotopy invariant, so that you can change $f$ into any homotopic map without changing $H_*f$. None of the properties you mentioned is homotopy invariant, so... –  Mariano Suárez-Alvarez Apr 25 '11 at 6:20
Mariano: I am asking if there are any specific cases that you can get any info about $f$. I am not interested in changing the spaces. In fact, changing a top space with a homotopy does not preserve all the structure of the original space. So this makes sense in particular situations(I think). –  niyazi Apr 25 '11 at 6:56
For instance, I would be very very happy if there is a theorem like,say,: "...In this particular case, growth of the homomorphism is this therefore Lipschitz constant is this..." –  niyazi Apr 25 '11 at 7:01
what Mariano is saying is precisely that you shouldn't be able to detect things like Lipschitz constants. –  Qiaochu Yuan Apr 25 '11 at 16:32
@Qiaochu: yes, I saw his point. But I am still not satisfied. He is saying that the conclusion should be homotopy invariant. Yes,so what? I am asking about what more you can deduce about $f$ in specific situations? I believe that for different topological spaces (other than spheres,CWs,...) more can be said about f. (BTW in the case of a selfmap of 1-point space you can detect being lipschitz :D ) –  niyazi Apr 26 '11 at 4:43

Any properties you listed are not homotopy invariant, therefore the homology functor is unable to detect these properties. Note that this does not mean that you take homotopies of spaces, just of functions. Any smooth function is homotopic to a continuous function which is not smooth, hence they induce the same group homomorphism. I think you can prove similar statements for the properties you listed.

There are of course many properties that the homology functor detects. A trivial example is the following: a function $S^1\rightarrow S^1$ that induces a non-trivial group homomorphism $\mathbb{Z}\rightarrow\mathbb{Z}$ has a non-contractible image.

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Yes, thank you. What are the nontrivial examples? Is there a degree theory for arbitrary spaces different than CW complexes and spheres? –  niyazi Apr 25 '11 at 11:55
I'm not sure about "different from CW-complexes" (pretty much anything you care about at least has the homotopy type of one), but anything with top-degree homology a $\mathbb{Z}$ has this notion of degree. In particular, any orientable manifold (and in that case the notion is more powerful because you also have Poincare duality to help you out.) –  Dylan Wilson Apr 25 '11 at 15:18