×

\(K\)-theoretic torsion and the zeta function. (English) Zbl 1493.19001

Let \(\tilde X \to X\) be a cyclic covering space with generating deck transformation \(T \colon \tilde X \to \tilde X\). Associated with the pair \((\tilde X, T)\) is the algebraic torsion \(\tau(t) \in \mathbb{Q}(t)^\times\) of the contractible chain complex \(C_*(\tilde X) \otimes_{\mathbb{Z}[t,t^{-1}]} \mathbb{Q}(t)\) (which has an interpretation of Reidemeister torsion of \(X\)), and the Lefschetz zeta function \[ \zeta(t) = \exp \Big( \sum_{k \geq 1} L(T^{\circ k} \frac{t^k}{k} \Big) \ \in \mathbb{Q}(t)^\times \ . \] Milnor showed that these two invariants are related to the Euler characteristic by the formula \[ \zeta(t^{-1}) \tau(t) = t^{\chi(\tilde X)} \ . \]
The authors construct algebraic versions of torsion, zeta function and Euler characteristic, giving maps \begin{gather*} \tau(t) \colon K(\mathrm{End}^S_A) \to \Omega K(A[t]_S) \ , \\ \zeta(t) \colon K(\mathrm{End}^S_A) \to \Omega K(A[t]_T) \ , \\ \chi(t) \colon K(\mathrm{End}^S_A) \to \Omega K(A[t]_S) \ ; \end{gather*} here \(A\) is an associative ring, \(S \subseteq A[t]\) is a set of central monic polynomials, \(T\) is obtained from \(S\) by “reversing” the coefficients, and the \(K\)-theory spaces are those associated with the category of \(S\)-torsion endomorphisms of \(A\)-modules and the indicated localisations of the ring \(A[t]\), respectively.
The main result is that the sum of \(\tau(t)\) and \(\zeta(t^{-1})\), in this new setting, is homotopic to the map \(\chi(t)\), generalising Milnor’s formula.
It is further explained how to use linearisation to extend the above constructions to categories of endomorphisms of retractive spaces (the input data for Waldhausen \(K\)-theory of spaces), giving higher \(K\)-theory versions of torsion and zeta function for spaces. The resulting invariants are shown to be non-trivial in higher \(K\)-groups, as shown by explicit examples.
The paper is carefully written and contains a number of interesting results apart from the ones mentioned above, including a “fundamental theorem” expressing the \(K\)-theory spaces \(K(A[t]_S)\) and \(K(A[t]_T)\) in terms of the \(K\)-theory spaces of \(A\), \(A[t]\) and the \(K\)-theory of \(S\)-torsion endomorphisms (with \(A\), \(S\) and \(T\) as before).

MSC:

19J10 Whitehead (and related) torsion
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.2140/gt.2013.17.733 · Zbl 1267.19001 · doi:10.2140/gt.2013.17.733
[2] ; Borel, Armand, Cohomologie réelle stable de groupes S-arithmétiques classiques, C. R. Acad. Sci. Paris Sér. A-B, 274 (1972) · Zbl 0235.57015
[3] 10.4310/HHA.2009.v11.n1.a2 · Zbl 1160.55005 · doi:10.4310/HHA.2009.v11.n1.a2
[4] 10.1007/BF02393236 · Zbl 1077.19002 · doi:10.1007/BF02393236
[5] 10.1007/BF01388745 · Zbl 0621.53035 · doi:10.1007/BF01388745
[6] ; Fried, David, Lefschetz formulas for flows, The Lefschetz centennial conference, Part III. Contemp. Math., 58, 19 (1987) · Zbl 0619.58034
[7] 10.1016/0040-9383(94)90004-3 · Zbl 0821.55001 · doi:10.1016/0040-9383(94)90004-3
[8] ; Gersten, S. M., Some exact sequences in the higher K-theory of rings, Algebraic K-theory, I : Higher K-theories. Lecture Notes in Math., 341, 211 (1973) · Zbl 0289.18011
[9] 10.1017/S002776300000221X · Zbl 0103.27001 · doi:10.1017/S002776300000221X
[10] 10.2307/1996180 · Zbl 0238.55001 · doi:10.2307/1996180
[11] ; Grayson, Daniel, Higher algebraic K-theory, II, Algebraic K-theory. Lecture Notes in Math., 551, 217 (1976) · Zbl 0362.18015
[12] 10.1016/0021-8693(77)90320-9 · Zbl 0413.18010 · doi:10.1016/0021-8693(77)90320-9
[13] ; Hausmann, Jean-Claude, Homology sphere bordism and Quillen plus construction, Algebraic K-theory. Lecture Notes in Math., 551, 170 (1976) · Zbl 0358.55005
[14] 10.1007/BF02392597 · Zbl 0892.19003 · doi:10.1007/BF02392597
[15] 10.1016/s0550-3213(02)00739-3 · Zbl 0999.81078 · doi:10.1016/s0550-3213(02)00739-3
[16] 10.4007/annals.2019.190.3.4 · Zbl 1427.19002 · doi:10.4007/annals.2019.190.3.4
[17] 10.4310/HHA.2016.v18.n1.a17 · Zbl 1387.19003 · doi:10.4310/HHA.2016.v18.n1.a17
[18] 10.1016/j.aim.2016.11.017 · Zbl 1360.19001 · doi:10.1016/j.aim.2016.11.017
[19] 10.1090/memo/0155 · Zbl 0321.55033 · doi:10.1090/memo/0155
[20] ; Milnor, John W., Infinite cyclic coverings, Conference on the topology of manifolds, 115 (1968) · Zbl 0179.52302
[21] 10.1007/BF01425486 · Zbl 0199.55501 · doi:10.1007/BF01425486
[22] ; Quillen, Daniel, Higher algebraic K-theory, I, Algebraic K-theory, I : Higher K-theories. Lecture Notes in Math., 341, 85 (1973) · Zbl 0292.18004
[23] ; Quillen, Daniel, Higher K-theory for categories with exact sequences, New developments in topology. London Math. Soc. Lecture Note Ser., 11, 95 (1974) · Zbl 0276.18013
[24] 10.1007/BFb0074449 · Zbl 0579.18006 · doi:10.1007/BFb0074449
[25] 10.1090/gsm/145 · doi:10.1090/gsm/145
[26] 10.1090/conm/126/00513 · Zbl 0780.19007 · doi:10.1090/conm/126/00513
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.