# What information can one recover from the induced map on homology?

The following question came up while constructing delay embeddings of time series data.

Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a combinatorial representation $S_X$ of $X$ via a finite simplicial complex and an unknown isomorphism $\gamma_*$ from the simplicial homology $\text{H}^\Delta_*(S_X)$ (which is computable) to the singular homology $\text{H}_*(X)$.

Similarly, instead of any direct knowledge of $f$, we have a simplicial map $\phi:S_X \to S_X$ and the commuting relation

$$f_* \circ \gamma_* \equiv \gamma_* \circ \phi_*$$

where the star subscripts indicate maps induced on homology groups.

So from the matrix representations of $\phi_*$ one can deduce how $f$ maps cycles in $X$. Perhaps more non-trivially, one could compute the Lefschetz number of $f$ via the alternating sum of traces formula and deduce the existence of fixed points. Here's my question:

What else can one infer about $f$ from the $\phi_*$ matrices?

Do the determinants, characteristic polynomials and other matrix invariants of the $\phi_*$'s also carry useful information about $f$?

Update: Since a clarification has been requested, here are the details of how one might construct $S_X$ from $X$. To begin with, $X \subset \mathbb{R}^n$ is a $k$-dimensional Riemannian submanifold of Euclidean space. One assumes the existence of a finite point set $P \subset \mathbb{R}^n$.

It is easy to see that if $P$ is sufficiently dense in $X$ then there is a radius $\epsilon > 0$ so that the union $U_\epsilon(P)$ of $n$-dimensional $\epsilon$-balls around points in $P$ covers $X$; in addition, if $\epsilon$ is small relative to the curvature of $X$, the map sending any point in $U_\epsilon(P)$ to its nearest point in $X$ is a strong deformation retraction, and so there is a homotopy equivalence between $U_\epsilon(P)$ and $X$.

In the question, $S_X$ is the Cech nerve of the obvious cover $\lbrace B_\epsilon(p)~|~p \in P\rbrace$ of $U_\epsilon(P)$. The map $\gamma_*$ comes from the fact that $S_X$ has the homology isomorphic to that of $U_\epsilon(P)$ by the nerve theorem, and that $U_\epsilon(P)$ in turn has the same homology as $X$ via the retraction outlined above. So in particular, there is no need to worry about which ring the homology coefficients come from: any PID will suffice.

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Hoping to clarify, in what sense is $X$ unknown, and in what sense is $S_X$ a "representation" of $X$? Do you mean that the only known connection is the given isomorphism $H(S) \simeq H(X)$? Is $|S_X|$ weak homotopy equivalent to $X$? Does the isomorphism work for all (constant) coefficients or only your-favourite-ring? – some guy on the street Jun 11 '12 at 18:01
Done! See the update in the question... – Vidit Nanda Jun 11 '12 at 18:55
One comment: since the matrices of $f_\ast$ and $\phi_\ast$ are similar, one can ask the same question without introducing $S_X$. (Probably one should then assume $H_\ast(X)$ is finitely generated in each degree.) – Mark Grant Jun 12 '12 at 6:10

When $X$ is a surface, then the moduli of the eigenvalues of $\phi_{\star}$ give lower bounds on the log of the dilatation of $f$, and therefore on the entropy of $f$. If we know additionally that the train track associated to $f$ is orientable, then the larger eigenvalue is the log of the dilatation. A more precise statement can be found in the following paper by Birman, Brinkman and Kawamuro http://arxiv.org/abs/1001.5094 where they associate several polynomials to $f$, one of them being actually associated to $\phi_\star$.
Pierre, could you please mention which statement from the Birman, Brinkman and Kawamuro paper you are referring to? I am slightly worried that their global assumption (requiring $f$ to be pseudo-Asonov and hence at least a homeomorphism) is perhaps too strong to be useful in the context of the question even when we restrict to surfaces. – Vidit Nanda Jun 11 '12 at 16:22