Geometry & Topology

On invariants of Hirzebruch and Cheeger–Gromov

Stanley Chang and Shmuel Weinberger

Full-text: Open access


We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that π1(M) 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 τ(2):S(M) that coincides with the ρ–invariant of Cheeger–Gromov. In particular, our result shows that the ρ–invariant is not a homotopy invariant for the manifolds in question.

Article information

Geom. Topol., Volume 7, Number 1 (2003), 311-319.

Received: 28 March 2003
Accepted: 30 April 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R67: Surgery obstructions, Wall groups [See also 19J25]
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 58G10

signature $L^2$–signature structure set $\rho$–invariant


Chang, Stanley; Weinberger, Shmuel. On invariants of Hirzebruch and Cheeger–Gromov. Geom. Topol. 7 (2003), no. 1, 311--319. doi:10.2140/gt.2003.7.311.

Export citation


  • M Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32–33 (1976) 43–72
  • M Atiyah, V Patodi, I Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • M Atiyah, V Patodi, I Singer, Spectral asymmetry and Riemannian geometry II, Math. Proc. Cambridge Philos. Soc. 78 (1975) 405–432
  • M Atiyah, V Patodi, I Singer, Spectral asymmetry and Riemannian geometry III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99
  • G Baumslag, E Dyer, A Heller, The topology of discrete groups, J. Pure Applied Alg. 16 (1980) 1–47
  • W Browder, R Livesay, Fixed point free involutions on homotopy spheres, Tôhoku Math. J. (2) 25 (1973) 69–87
  • J Cheeger, M Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential Geometry and Complex Analysis, 115–154, Springer, Berlin–New York (1985)
  • 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) 1–34
  • P E Conner, E E Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 33, Academic Press Inc. New York; Springer–Verlag, Berlin, (1964)
  • F T Farrell, L E Jones, Rigidity for aspherical manifolds with $\pi_ 1\subset{\rm GL}_m(\reals)$, Asian J. Math. 2 (1998) 215–262
  • J C Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981) 115–123
  • 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) 131–142
  • F Hirzebruch, Involutionen auf Mannigfaltigkeiten (German) 1968 Proc. Conf. on Transformation Groups (New Orleans, LA 1967) 148–166
  • N Keswani, Homotopy invariance of relative eta-invariants and $C^\ast$–algebra $K$–theory, Electron. Res. Announc. Amer. Math. Soc. 4 (1998) 18–26
  • S Lopez de Medrano, Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 59, Springer–Verlag, New York–Heidelberg (1971)
  • W Lück, $L^2$–invariants of regular coverings of compact manifolds and CW–complexes, to appear in the Handbook of Geometric Topology
  • W Lück, $L^2$–Invariants: Theory and Applications to Geometry and K–Theory. A Series of Modern Surveys in Mathematics, vol. 44, Springer–Verlag, Berlin (2002)
  • W Lück, T Schick, Various $L^2$–signatures and a topological $L^2$–signature theorem, preprint
  • W Lück, T Schick, Approximating $L^2$–signatures by their finite-dimensional analogues, Preprintreihe SFB 478 – Geometrische Strukturen in der Mathematik, Heft 190, Münster (2001)
  • W Mathai, Spectral flow, eta-invariants and von Neumann algebras, J. Funct. Anal. 109 (1992) 442–456
  • M Ramachandran, von Neumann index theorems for manifolds with boundary, J. Differential Geom. 38 (1993) 315–349
  • 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, Berlin(1979) 102–157,
  • B Vaillant, Indextheorie für Überlangerungen, Diplomarbeit, Universität Bonn, (1997)
  • C T C Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London–New York (1970)
  • S Weinberger, Homotopy invariance of $\eta$–invariants, Proc. Nat. Acad. Sci. USA 83 (1988) 5362–5363 \bignohang [We2] S Weinberger, The Topological Classification of Stratified Spaces, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1994)