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A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic (co)chain complex of vector bundles) are discussed. Namely, let $E\to X$ be a vector bundle with finite-dimensional fibres, every of which is isomorphic to a graded space $C=\bigoplus_{i=0}^n C_i$. Also assume, that a (smooth) family of (co)chain differentials is given (one for every point in $X$), making every fibre into an acyclic (co)chain complex $C_n\to C_{n-1}\to\dots\to C_0$. Now the question is, if there exists a parametrised version of Reidemeister torsion (or some higher versions of this invariant) in this case? If you know a paper or a book, in which such questions are discussed, please, let me know!

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  • $\begingroup$ I'm not sure I understand your question, but a possible answer is Kiyoshi Igusa's book "Higher Franz-Reidemeister Torsion." I should mention analogously parameterized version of Whitehead torsion is more popular, under the name the "Whitehead space." $\endgroup$ Oct 9, 2011 at 17:41
  • $\begingroup$ Thank you. In effect, it was from reading this book that I came to asking such questions... The main problem with Igusa's construction is the use of the two-index theorem, which is very indirect, so one cannot get an explicit formula from it. $\endgroup$
    – gshar
    Oct 9, 2011 at 18:12

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