I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now suppose for any finite-dimensional simplex $K$ I have the $i$-th Homology group with rational coefficients, $H_i(K,\mathbb{Q})$ and $f:K\rightarrow K$. Then we have the induced map $f_*:H_i(K,\mathbb{Q})\rightarrow H_i(K,\mathbb{Q})$. Since $f_*$ is a map from a finite-dimensional vetor space to itself, we can talk about its matrix and the alternating sum of the trace of the matrices of $i$-th Homology groups give the Lefschetz's Number for $f$.

1) I was wondering that what could be the $motivation/insight$ behind using the trace of the matrix of $f_*$ for various purposes.

2) Again, for any matrix, its trace as well as determinant are invariants, so I was wondering that are there any instances where we use the determinant of $f_*$ for associating something like Lefschetz's number. Is there any example in the literature where the determinant of $f_*$ serves any purpose or is this just a vague idea.

I tried to think about the 2nd question by taking some simplicial maps for lower-dimensional simplexes, but couldn't get anything promising. I searched about it on the internet, but couldn't find anything substantial.

  • 1
    $\begingroup$ For (1): The trace of the matrix associated to a permutation of the basis of your vector space is precisely the number of fixed points of the permutation. For (2): Don't have much but, if you have a self-map of a torus T^n then the determinant of the map on H^1 is the map on H^n. $\endgroup$ – Dylan Wilson Nov 8 '14 at 16:06
  • $\begingroup$ Determinants of complexes and of maps of complexes (and these are 'alternating products") are useful in several contexts. See the book of Gelfand, Kapranov and Zelevinski for information, for example. $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '14 at 18:40
  • $\begingroup$ @MarianoSuárez-Alvarez Thanks for the references...i'll have a look at them. But what does one mean by determinant of complexes ? $\endgroup$ – wanderer Nov 8 '14 at 20:43
  • $\begingroup$ Well, that is very greatly explained in GKZ :-) $\endgroup$ – Mariano Suárez-Álvarez Nov 8 '14 at 20:51
  • $\begingroup$ I know a role that the determinant plays not in the homology of topological spaces but in the cohomology of algebraic varieties: the rationality part of the Weil conjectures is proven by showing that the zeta function is given by the alternating product of the characteristic polynomials of the maps the Frobenius morphism induces on the cohomology. $\endgroup$ – Tom Price Nov 9 '14 at 1:25

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