# 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

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.
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