The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds

A. A. Gaifullin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 44-63.

Résumé

We consider the classical N. Steenrod's problem of realization of cycles by continuous images of manifolds. Our goal is to find a class $\mathcal M_n$ of oriented $n$-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image of a manifold from the class $\mathcal M_n$. We prove that as the class $\mathcal M_n$ one can take a set of finite-fold coverings of the manifold $M^n$ of isospectral symmetric tridiagonal real $(n+1)\times(n+1)$ matrices. It is well known that the manifold $M^n$ is aspherical, its fundamental group is torsion-free, and its universal covering is diffeomorphic to $\mathbb R^n$. Thus, every integral homology class of an arcwise connected space can be realized with some multiplicity by an image of an aspherical manifold with a torsion-free fundamental group. In particular, for any closed oriented manifold $Q^n$, there exists an aspherical manifold that has torsion-free fundamental group and can be mapped onto $Q^n$ with nonzero degree.

Bibliographie

[1] Burbaki N., Gruppy i algebry Li, Glavy 4–6, Mir, M., 1972 | MR | Zbl

[2] Bukhshtaber V. M., “Moduli differentsialov spektralnoi posledovatelnosti Atya–Khirtsebrukha. I”, Mat. sb., 78(120):2 (1969), 307–320 ; “II”, 83(125):1 (1970), 61–76 | MR | Zbl | MR | Zbl

[3] Bukhshtaber V. M., Panov T. E., Toricheskie deistviya v topologii i kombinatorike, MTsNMO, M., 2004 | MR

[4] Vinberg E. B., “Diskretnye lineinye gruppy, porozhdennye otrazheniyami”, Izv. AN SSSR. Ser. mat., 35:5 (1971), 1072–1112 | MR | Zbl

[5] Gaifullin A. A., “Lokalnye formuly dlya kombinatornykh klassov Pontryagina”, Izv. RAN. Ser. mat., 68:5 (2004), 13–66 | MR | Zbl

[6] Gaifullin A. A., “Yavnoe postroenie mnogoobrazii, realizuyuschikh zadannye klassy gomologii”, UMN, 62:6 (2007), 167–168 | MR | Zbl

[7] Gaifullin A. A., “Realizatsiya tsiklov asferichnymi mnogoobraziyami”, UMN, 63:3 (2008), 157–158 | MR | Zbl

[8] Novikov S. P., “Gomotopicheskie svoistva kompleksov Toma”, Mat. sb., 57(99):4 (1962), 407–442 | MR | Zbl

[9] Tom R., “Nekotorye svoistva “v tselom” differentsiruemykh mnogoobrazii”, Rassloennye prostranstva i ikh prilozheniya, Izd-vo inostr. lit., M., 1958, 291–348 | MR

[10] Davis M. W., “Groups generated by reflections and aspherical manifolds not covered by Euclidean space”, Ann. Math. Ser. 2, 117:2 (1983), 293–324 | DOI | MR | Zbl

[11] Davis M. W., “Some aspherical manifolds”, Duke Math. J., 55:1 (1987), 105–139 | DOI | MR | Zbl

[12] Davis M. W., Januszkiewicz T., “Convex polytopes, Coxeter orbifolds and torus actions”, Duke Math. J., 62:2 (1991), 417–451 | DOI | MR | Zbl

[13] Eilenberg S., “On the problems of topology”, Ann. Math. Ser. 2, 50 (1949), 247–260 | DOI | MR | Zbl

[14] Ferri M., “Una rappresentazione delle $n$-varietà topologiche triangolabili mediante grafi $(n+1)$-colorati”, Boll. Un. Mat. Ital. B (5), 13:1 (1976), 250–260 | MR | Zbl

[15] Ferri M., Gagliardi C., Grasselli L., “A graph-theoretical representation of PL-manifolds – a survey on crystallizations”, Aequat. Math., 31:2–3 (1986), 121–141 | DOI | MR | Zbl

[16] Pezzana M., “Diagrammi di Heegaard e triangolazione contratta”, Boll. Un. Mat. Ital. (4), 12:3, Suppl. (1975), 98–105 | MR | Zbl

[17] Sullivan D., “Singularities in spaces”, Proc. Liverpool Singularities Symposium, II, Lect. Notes Math., 209, Springer, Berlin, 1971, 196–206 | MR

[18] Tomei C., “The topology of isospectral manifolds of tridiagonal matrices”, Duke Math. J., 51:4 (1984), 981–996 | DOI | MR | Zbl

[19] Ziegler G. M., Lectures on polytopes, Grad. Texts Math., 152, Springer, Berlin, 1995 | MR | Zbl