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The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture

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Abstract

An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric L-spectrum of \({\mathbb {Z}}\), which is, up to weak equivalence, an \(E_\infty \) ring map. Using this map, we construct a fundamental L-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces whose fundamental group satisfies the Novikov conjecture.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Heidelberg, INF 288, 69120, Heidelberg, Germany

    Markus Banagl

  2. Fakultät für Mathematik, Ruhr-Universität Bochum, IB 3/179-Fach 41, 44780, Bochum, Germany

    Gerd Laures

  3. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    James E. McClure

Authors
  1. Markus Banagl
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  2. Gerd Laures
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  3. James E. McClure
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Correspondence to Gerd Laures.

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M. Banagl was supported in part by a research grant of the Deutsche Forschungsgemeinschaft. J.E. McClure was partially supported by a grant from the Simons Foundation (#279092 to James McClure). He thanks the Lord for making his work possible.

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Banagl, M., Laures, G. & McClure, J.E. The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture. Sel. Math. New Ser. 25, 7 (2019). https://doi.org/10.1007/s00029-019-0458-y

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