Geometry & Topology

On the topological contents of $\eta$–invariants

Ulrich Bunke

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We discuss a universal bordism invariant obtained from the Atiyah–Patodi–Singer η–invariant from the analytic and homotopy-theoretic point of view. Classical invariants like the Adams e–invariant, ρ–invariants and String–bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.

Article information

Geom. Topol., Volume 21, Number 3 (2017), 1285-1385.

Received: 8 November 2013
Revised: 20 May 2016
Accepted: 5 September 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J28: Eta-invariants, Chern-Simons invariants

eta invariant $K$–theory bordism


Bunke, Ulrich. On the topological contents of $\eta$–invariants. Geom. Topol. 21 (2017), no. 3, 1285--1385. doi:10.2140/gt.2017.21.1285.

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  • J F Adams, On the groups $J(X)$, IV, Topology 5 (1966) 21–71
  • J F Adams, Stable homotopy and generalised homology, Univ. Chicago Press (1974)
  • J F Adams, Infinite loop spaces, Annals of Mathematics Studies 90, Princeton Univ. Press (1978)
  • J F Adams, A S Harris, R M Switzer, Hopf algebras of cooperations for real and complex $K$–theory, Proc. London Math. Soc. 23 (1971) 385–408
  • D W Anderson, L Hodgkin, The $K$–theory of Eilenberg–MacLane complexes, Topology 7 (1968) 317–329
  • M A Ando, M I Hopkins, C Rezk, Multiplicative orientations of $\mathrm{KO}$–theory and of the spectrum of topological modular forms, preprint (2010) Available at \setbox0\makeatletter\@url {\unhbox0
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, II, Math. Proc. Cambridge Philos. Soc. 78 (1975) 405–432
  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99
  • M F Atiyah, G B Segal, Equivariant $K$–theory and completion, J. Differential Geometry 3 (1969) 1–18
  • M F Atiyah, I M Singer, The index of elliptic operators, I, Ann. of Math. 87 (1968) 484–530
  • A Bahri, P Gilkey, The eta invariant, ${\rm Pin}^c$ bordism, and equivariant ${\rm Spin}^c$ bordism for cyclic $2$–groups, Pacific J. Math. 128 (1987) 1–24
  • A Bahri, P Gilkey, ${\rm Pin}^c$ cobordism and equivariant ${\rm Spin}^c$ cobordism of cyclic $2$–groups, Proc. Amer. Math. Soc. 99 (1987) 380–382
  • T Bauer, Computation of the homotopy of the spectrum tmf, from “Groups, homotopy and configuration spaces” (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13, Geom. Topol. Publ., Coventry (2008) 11–40
  • P Baum, R G Douglas, $K$ homology and index theory, from “Operator algebras and applications, I” (R V Kadison, editor), Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, RI (1982) 117–173
  • M Behrens, G Laures, $\beta$–family congruences and the $f$–invariant, from “New topological contexts for Galois theory and algebraic geometry” (A Baker, B Richter, editors), Geom. Topol. Monogr. 16, Geom. Topol. Publ., Coventry (2009) 9–29
  • N Berline, E Getzler, M Vergne, Heat kernels and Dirac operators, Grundl. Math. Wissen. 298, Springer, Berlin (1992) Corrected reprint 2004
  • B Blackadar, $K$–theory for operator algebras, 2nd edition, Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press (1998)
  • J M Boardman, Stable operations in generalized cohomology, from “Handbook of algebraic topology” (I M James, editor), North-Holland, Amsterdam (1995) 585–686
  • A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
  • J-L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics 107, Birkhäuser, Boston (1993)
  • U Bunke, A $K$–theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995) 241–279
  • U Bunke, Index theory, eta forms, and Deligne cohomology, Mem. Amer. Math. Soc. 928, Amer. Math. Soc., Providence, RI (2009)
  • U Bunke, N Naumann, Secondary invariants for string bordism and topological modular forms, Bull. Sci. Math. 138 (2014) 912–970
  • U Bunke, T Schick, Smooth $K$–theory, from “From probability to geometry, II” (X Dai, R Léandre, X Ma, W Zhang, editors), Astérisque 328, Soc. Math. France, Paris (2009) 45–135
  • U Bunke, T Schick, Uniqueness of smooth extensions of generalized cohomology theories, J. Topol. 3 (2010) 110–156
  • D Crowley, S Goette, Kreck–Stolz invariants for quaternionic line bundles, Trans. Amer. Math. Soc. 365 (2013) 3193–3225
  • P Deligne, Cohomologie étale (SGA $4\chalf\kern-.1em$), Lecture Notes in Math. 569, Springer, Berlin (1977)
  • C Deninger, W Singhof, The $e$–invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. Math. 78 (1984) 101–112
  • H Donnelly, Spectral geometry and invariants from differential topology, Bull. London Math. Soc. 7 (1975) 147–150
  • A Z Dymov, Homology spheres and contractible compact manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 72–77 In Russian; translated in Math. USSR-Izv. 5 (1971) 73–79
  • D S Freed, M Hopkins, On Ramond–Ramond fields and $K$–theory, J. High Energy Phys. (2000) art. id. 44
  • D S Freed, R B Melrose, A mod $k$ index theorem, Invent. Math. 107 (1992) 283–299
  • B I Gray, Spaces of the same $n$–type, for all $n$, Topology 5 (1966) 241–243
  • F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Mathematics E20, Friedr. Vieweg & Sohn, Braunschweig (1992)
  • N Hitchin, Lectures on special Lagrangian submanifolds, from “Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds” (C Vafa, S-T Yau, editors), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, RI (2001) 151–182
  • M J Hopkins, Algebraic topology and modular forms, from “Proceedings of the International Congress of Mathematicians, I: Plenary lectures and ceremonies” (T Li, editor), Higher Ed. Press, Beijing (2002) 291–317
  • M J Hopkins, I M Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005) 329–452
  • M A Hovey, $v_n$–elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997) 327–356
  • M Hovey, The homotopy of $M\rm String$ and $M\rm U\langle 6\rangle$ at large primes, Algebr. Geom. Topol. 8 (2008) 2401–2414
  • M A Hovey, D C Ravenel, The $7$–connected cobordism ring at $p=3$, Trans. Amer. Math. Soc. 347 (1995) 3473–3502
  • J D S Jones, B W Westbury, Algebraic $K$–theory, homology spheres, and the $\eta$–invariant, Topology 34 (1995) 929–957
  • G G Kasparov, Equivariant $KK$–theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201
  • M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
  • M Kreck, S Stolz, A diffeomorphism classification of $7$–dimensional homogeneous Einstein manifolds with ${\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$–symmetry, Ann. of Math. 127 (1988) 373–388
  • G Laures, The topological $q$–expansion principle, PhD thesis, Massachusetts Institute of Technology (1996) Available at \setbox0\makeatletter\@url {\unhbox0
  • G Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000) 5667–5688
  • H B Lawson, Jr, M-L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton Univ. Press (1989)
  • J Lurie, A survey of elliptic cohomology, from “Algebraic topology” (N A Baas, E M Friedlander, B Jahren, P A Østvær, editors), Abel Symp. 4, Springer, Berlin (2009) 219–277
  • E Y Miller, R Lee, Some invariants of spin manifolds, Topology Appl. 25 (1987) 301–311
  • H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. of Math. 106 (1977) 469–516
  • M K Murray, An introduction to bundle gerbes, from “The many facets of geometry” (O García-Prada, J P Bourguignon, S Salamon, editors), Oxford Univ. Press (2010) 237–260
  • D Quillen, Letter from Quillen to Milnor on ${\rm Im}(\pi \sb{i}O\rightarrow \pi \sb{i}\sp{{\rm s}}\rightarrow K\sb{i}{\bf Z})$, from “Algebraic $K$–theory” (M R Stein, editor), Lecture Notes in Math. 551, Springer, Berlin (1976) 182–188
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press, Orlando, FL (1986)
  • Y B Rudyak, On Thom spectra, orientability, and cobordism, Springer, Berlin (1998)
  • J A Seade, On the $\eta $–function of the Dirac operator on $\Gamma \backslash S\sp{3}$, An. Inst. Mat. Univ. Nac. Autónoma México 21 (1981) 129–147
  • J-P Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math. 58 (1953) 258–294
  • A A Suslin, On the $K$–theory of local fields, J. Pure Appl. Algebra 34 (1984) 301–318
  • M Völkl, Universal geometrizations and the intrinsic eta-invariant, PhD thesis, Universität Regensburg (2015) Available at \setbox0\makeatletter\@url {\unhbox0
  • K Waldorf, String connections and Chern–Simons theory, Trans. Amer. Math. Soc. 365 (2013) 4393–4432