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Keywords:
de Rham cohomology; Lefschetz coincidence number; solvmanifold
Summary:
For any two continuous maps $f$, $g$ between two solvmanifolds of the same dimension satisfying the Mostow condition, we give a technique of computation of the Lefschetz coincidence number of $f$, $g$. This result is an extension of the result of Ha, Lee and Penninckx for completely solvable case.
References:
[1] Auslander, L.: An exposition of the structure of solvmanifolds. I. Algebraic theory. Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. DOI 10.1090/S0002-9904-1973-13134-9 | MR 0486307 | Zbl 0265.22016
[2] Baues, O., Klopsch, B.: Deformations and rigidity of lattices in solvable Lie groups. J. Topol. (online published). MR 3145141
[3] Console, S., Fino, A.: On the de Rham cohomology of solvmanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 10 (2011), no. 4, 801–811. MR 2932894 | Zbl 1242.53055
[4] Ha, K.Y., Lee, J.B., Penninckx, P.: Anosov theorem for coincidences on special solvmanifolds of type (R). Proc. Amer. Math. Soc. 139 (2011), no. 6, 2239–2248. MR 2775401
[5] Hattori, A.: Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331. MR 0124918 | Zbl 0099.18003
[6] Jezierski, J., Marzantowicz, W.: Homotopy methods in topological fixed and periodic points theory. Topol. Fixed Point Theory Appl., vol. 3, Springer, Dordrecht, 2006. MR 2189944 | Zbl 1085.55001
[7] Kasuya, H.: The Frolicher spectral sequences of certain solvmanifolds. J. Geom. Anal. (2013), Online First. DOI 10.1007/s12220-013-9429-2 | MR 3299283
[8] Kasuya, H.: Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds. Bull. Lond. Math. Soc. 45 (2013), no. 1, 15–26. DOI 10.1112/blms/bds057 | MR 3033950 | Zbl 1262.53061
[9] McCleary, J.: A user’s guide to spectral sequences. second ed., Cambridge Studies in Advanced Mathematics, Cambridge, 2001. MR 1793722 | Zbl 0959.55001
[10] McCord, C. K.: Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds. Topology Appl. 75 (1997), no. 1, 81–92. DOI 10.1016/S0166-8641(96)00081-8 | MR 1425386 | Zbl 1001.55004
[11] McCord, C.K., : Nielsen numbers and Lefschetz numbers on solvmanifolds. Pacific J. Math. 147 (1991), no. 1, 153–164. DOI 10.2140/pjm.1991.147.153 | MR 1081679 | Zbl 0666.55002
[12] Mostow, G.D.: Cohomology of topological groups and solvmanifolds. Ann. of Math. (2) 73 (1961), 20–48. MR 0125179 | Zbl 0103.26501
[13] Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. (2) 59 (1954), 531–538. DOI 10.2307/1969716 | MR 0064057 | Zbl 0058.02202
[14] Onishchik, A.L., Vinberg, E.B.: Lie groups and Lie algebras II. Springer, 2000. MR 1756406 | Zbl 0932.00011
[15] Raghnathan, M.S.: Discrete subgroups of Lie Groups. Springer-Verlag, New York, 1972. MR 0507234
[16] Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, 1951. MR 0039258 | Zbl 0054.07103
[17] Witte, D.: Superrigidity of lattices in solvable Lie groups. Invent. Math. 122 (1995), no. 1, 147–193. DOI 10.1007/BF01231442 | MR 1354957 | Zbl 0844.22015
[18] Wong, P.: Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds. J. Reine Angew. Math. 524 (2000), 185–204. MR 1770607 | Zbl 0962.55002
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