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