On invariants of Hirzebruch and Cheeger–Gromov

Chang, Stanley¹ ; Weinberger, Shmuel²

¹ Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02481, USA
² Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA

Geometry & topology, Tome 7 (2003) no. 1, pp. 311-319.

Résumé

We prove that, if $M$ is a compact oriented manifold of dimension $4k+3$, where $k>0$, such that ${\pi }_1\left(M\right)$ is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to $M$ but not homeomorphic to it. To show the infinite size of the structure set of $M$, we construct a secondary invariant ${\tau }_{\left(2\right)}:S\left(M\right)\to ℝ$ that coincides with the $\rho$–invariant of Cheeger–Gromov. In particular, our result shows that the $\rho$–invariant is not a homotopy invariant for the manifolds in question.

Détail

DOI : 10.2140/gt.2003.7.311

Keywords: signature, $L^2$–signature, structure set, $\rho$–invariant

Bibliographie

[1] M F Atiyah, Elliptic operators, discrete groups and von Neumann algebras, from: "Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan (Orsay, 1974)", Soc. Math. France (1976)

[2] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43

[3] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambridge Philos. Soc. 78 (1975) 405

[4] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71

[5] G Baumslag, E Dyer, A Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980) 1

[6] W Browder, G R Livesay, Fixed point free involutions on homotopy spheres, Tôhoku Math. J. $(2)$ 25 (1973) 69

[7] J Cheeger, M Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, from: "Differential geometry and complex analysis", Springer (1985) 115

[8] J Cheeger, M Gromov, Bounds on the von Neumann dimension of $L^2$–cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1

[9] P E Conner, E E Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33, Academic Press (1964)

[10] F T Farrell, L E Jones, Rigidity for aspherical manifolds with $\pi_1{\subset}\mathrm{GL}_m(\mathbb{R})$, Asian J. Math. 2 (1998) 215

[11] J C Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981) 115

[12] N Higson, G Kasparov, Operator $K$–theory for groups which act properly and isometrically on Hilbert space, Electron. Res. Announc. Amer. Math. Soc. 3 (1997) 131

[13] F Hirzebruch, Involutionen auf Mannigfaltigkeiten, from: "Proc. Conf. on Transformation Groups (New Orleans, La., 1967)", Springer (1968) 148

[14] N Keswani, Homotopy invariance of relative eta-invariants and $C^*$–algebra $K$–theory, Electron. Res. Announc. Amer. Math. Soc. 4 (1998) 18

[15] S López De Medrano, Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 59, Springer (1971)

[16] W Lück, $L^2$–invariants of regular coverings of compact manifolds and CW–complexes, from: "Handbook of geometric topology", North-Holland (2002) 735

[17] W Lück, $L^2$–invariants: theory and applications to geometry and $K$–theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer (2002)

[18] W Lück, T Schick, Various $L^2$–signatures and a topological $L^2$–signature theorem, from: "High-dimensional manifold topology", World Sci. Publ., River Edge, NJ (2003) 362

[19] W Lück, T Schick, Approximating $L^2$–signatures by their compact analogues, Forum Math. 17 (2005) 31

[20] V Mathai, Spectral flow, eta invariants, and von Neumann algebras, J. Funct. Anal. 109 (1992) 442

[21] M Ramachandran, von Neumann index theorems for manifolds with boundary, J. Differential Geom. 38 (1993) 315

[22] A Ranicki, Localization in quadratic $L$–theory, from: "Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978)", Lecture Notes in Math. 741, Springer (1979) 102

[23] B Vaillant, Indextheorie für Überlangerungen, Diplomarbeit, Universität Bonn (1997)

[24] C T C Wall, Surgery on compact manifolds, London Mathematical Society Monographs 1, Academic Press (1970)

[25] S Weinberger, Homotopy invariance of $\eta$–invariants, Proc. Nat. Acad. Sci. U.S.A. 85 (1988) 5362