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Twisted Lefschetz numbers of infra-solvmanifolds and algebraic groups. (English) Zbl 1432.22014
Let \(M\) be an infra-solvmanifold with the fundamental group \(\Gamma\) and \(\varphi:M\to M\) be a diffeomorphism. Denote by \(f:\Gamma\to \Gamma\) the homomorphism induced by \(\phi\). For \(V_\phi\in\mathrm{Rep}(\Gamma)\), we define the flat bundle \(E_\phi=(\widetilde M\times V_\phi)/\Gamma\). Take \(V_\phi\in\mathrm{Rep}(\mathcal{A}_\Gamma/\mathcal{U}_\Gamma)\) and an isomorphism \(\Psi:V_{\phi\circ F}\to V_\phi\). Considering \(\Psi:V_{\phi\circ f}\to V_\phi\) as an isomorphism in \(\mathrm{Rep}(\Gamma)\), \(\Psi\) corresponds to an isomorphism \(\Xi:\varphi^*E_\phi\to E_\phi\) of flat bundles.
The following is the main result in this paper:
Theorem. Let \(A\) be a matrix representation of \(F_{\mathcal{U}_{\Gamma^*}}:u\to u\). Then, necessarily, \(L(\varphi,E_\phi,\Xi)=\mathrm{det}(I-A)\).
Further aspects occasioned by these developments are also discussed.
22E25 Nilpotent and solvable Lie groups
54H25 Fixed-point and coincidence theorems (topological aspects)
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[1] Atiyah, M. F.; Bott, R., A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2), 88, 451-491 (1968) · Zbl 0167.21703
[2] Baues, O., Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43, 4, 903-924 (2004) · Zbl 1059.57022
[3] Brown, K., Cohomology of Groups (1982), Springer: Springer Berlin, Heidelberg, New York
[4] Farrell, F. T.; Jones, L. E., Classical Aspherical Manifolds (1990), Amer. Math. Soc.
[5] Hain, R., Remarks on non-abelian cohomology of proalgebraic groups, J. Algebraic Geom., 22, 3, 581-598 (2013) · Zbl 1312.14071
[6] Hattori, A., Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci., Univ. Tokyo, Sect. I, 8, 289-331 (1960) · Zbl 0099.18003
[7] Hochschild, G., Cohomology of algebraic linear groups, Illinois J. Math., 5, 492-519 (1961) · Zbl 0103.26502
[8] Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, vol. 131 (1987), Academic Press, Inc.: Academic Press, Inc. Boston, MA · Zbl 0652.20042
[9] Jezierski, J.; Marzantowicz, W., Homotopy Methods in Topological Fixed and Periodic Points Theory, Topological Fixed Point Theory and Its Applications, vol. 3 (2006), Springer: Springer Dordrecht · Zbl 1085.55001
[10] Kasuya, H., Central theorems for cohomologies of certain solvable groups, Trans. Amer. Math. Soc., 369, 4, 2879-2896 (2017) · Zbl 1401.20035
[11] Malcev, A., On a class of homogeneous spaces, Izv. Akad. Nauk SSSR. Ser. Mat., 13, 9-32 (1949), (Russian)
[12] McCord, C. K., Nielsen numbers and Lefschetz numbers on solvmanifolds, Pacific J. Math., 147, 1, 153-164 (1991) · Zbl 0666.55002
[13] Mostow, G. D., Fully reducible subgroups of algebraic groups, Amer. J. Math., 78, 200-221 (1956) · Zbl 0073.01603
[14] Nomizu, K., On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2), 59, 531-538 (1954) · Zbl 0058.02202
[15] (Onishchik, A. L.; Vinberg, E. B., Lie Groups and Lie Algebras II (2000), Springer) · Zbl 0932.00011
[16] Raghunathan, M. S., Discrete Subgroups of Lie Groups (1972), Springer-Verlag: Springer-Verlag New York · Zbl 0254.22005
[17] Tauvel, P.; Yu, R. W.T., Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1068.17001
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