Nitsche, Martin; Schick, Thomas; Zeidler, Rudolf

Transfer Maps in Generalized Group Homology via Submanifolds

Doc. Math. 26, 947-979 (2021)
DOI: 10.25537/dm.2021v26.947-979
Communicated by Mike Hill

Summary

Let \(N \subset M\) be a submanifold embedding of spin manifolds of some codimension \(k \geq 1\). A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that \(M\) does not admit a metric of positive scalar curvature if \(k = 2\) and the Dirac operator of \(N\) has non-trivial index, provided that suitable geometric conditions on \(N \subset M\) are satisfied. In the cases \(k=1\) and \(k=2\), Zeidler and Kubota, respectively, established more systematic results: There exists a transfer \(\text{KO}_\ast(\text{C}^{\ast} \pi_1 M)\to \text{KO}_{\ast - k}(\text{C}^\ast \pi_1 N)\) which maps the index class of \(M\) to the index class of \(N\). The main goal of this article is to construct analogous transfer maps \(E_\ast(\text{B}\pi_1M) \to E_{\ast-k}(\text{B}\pi_1N)\) for different generalized homology theories \(E\) and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer \(E_\ast(M) \to E_{\ast-k}(N)\) induced by the inclusion \(N \subset M\) for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for \(k \leq 3\). For \(k \leq 2\), we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Mathematics Subject Classification

55N20, 55N22, 55N91, 19K35

Keywords/Phrases

transfer maps, geneneralized cohomology, group cohomology, codimension 2 submanifold obstruction to positive scalar curvature

References

  • 1. Adams, J. F., Stable homotopy and generalised homology (1974), University of Chicago Press, Chicago, Ill.-London; zbl 0309.55016; MR0402720.
  • 2. Schick, Thomas; Higson, Nigel; Baum, Paul, A geometric description of equivariant \(K\)-homology for proper actions. In: Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, Quanta of Maths; zbl 1216.19006; MR2732043; arxiv 0907.2066.
  • 3. Blackadar, Bruce, \(K\)-theory for operator algebras, Mathematical Sciences Research Institute Publications, 5 (1998), Cambridge University Press, Cambridge; zbl 0913.46054; MR1656031.
  • 4. Bredon, Glen E., Sheaf theory (1967), McGraw-Hill Book Co., New York-Toronto, Ont.-London; zbl 0158.20505; MR0221500.
  • 5. Echterhoff, Siegfried; Chabert, Jérôme, Permanence properties of the Baum-Connes conjecture, Doc. Math., 6, 127-183 (2001); zbl 0984.46047; MR1836047; https://www.elibm.org/article/10000526.
  • 6. Engel, Alexander, Wrong way maps in uniformly finite homology and homology of groups, J. Homotopy Relat. Struct., 13, 2, 423-441 (2018); DOI 10.1007/s40062-017-0187-x; zbl 1401.55006; MR3802801; arxiv 1602.03374.
  • 7. Engel, Alexander, Correction to: Wrong way maps in uniformly finite homology and homology of groups, J. Homotopy Relat. Struct., 14, 4, 1143-1144 (2019); DOI 10.1007/s40062-019-00246-z; zbl 1447.55008; MR4025601.
  • 8. Lawson, H. Blaine, Jr.; Gromov, Mikhael, Spin and scalar curvature in the presence of a fundamental group. I, Ann. of Math. (2), 111, 2, 209-230 (1980); DOI 10.2307/1971198; zbl 0445.53025; MR0569070.
  • 9. Schick, Thomas; Pape, Daniel; Hanke, Bernhard, Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble), 65, 6, 2681-2710 (2015); DOI 10.5802/aif.3000; zbl 1344.58012; MR3449594; arxiv 1402.4094.
  • 10. Kasparov, G. G., \(K\)-theory, group \(C^*\)-algebras, and higher signatures (conspectus). In: London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, Novikov conjectures, index theorems and rigidity, Vol. 1 (1993); DOI 10.1017/CBO9780511662676.007; zbl 0957.58020; MR1388299.
  • 11. Kubota, Yosuke, The relative Mishchenko-Fomenko higher index and almost flat bundles. II: Almost flat index pairing (Preprint, 2019); arxiv 1908.10733.
  • 12. Schick, Thomas; Kubota, Yosuke, The Gromov-Lawson codimension 2 obstruction to positive scalar curvature and the \(C^*\)-index, Geom. Topol., 25, 2, 949-960 (2021); DOI 10.2140/gt.2021.25.949; zbl 07343536; MR4251439; arxiv 1909.09584.
  • 13. Lurie, Jacob, Chromatic Homotopy Theory; http://www.math.harvard.edu/~lurie/252x.html.
  • 14. Lück, Wolfgang, Chern characters for proper equivariant homology theories and applications to \(K\)- and \(L\)-theory, J. Reine Angew. Math., 543, 193-234 (2002); DOI 10.1515/crll.2002.015; zbl 0987.55008; MR1887884.
  • 15. Mislin, Guido, Equivariant \(K\)-homology of the classifying space for proper actions. In: Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, Proper group actions and the Baum-Connes conjecture; MR2027169.
  • 16. Nitsche, Martin, New topological and index-theoretical methods to study the geometry of manifolds (2018), University of Göttingen; zbl 1409.58001.
  • 17. Oyono-Oyono, Hervé, Baum-Connes conjecture and group actions on trees, \(K\)-Theory, 24, 2, 115-134 (2001); DOI 10.1023/A:1012786413219; zbl 1008.19001; MR1869625.
  • 18. Rudyak, Yuli B., On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematics (1998), Springer-Verlag, Berlin; DOI 10.1007/978-3-540-77751-9; zbl 0906.55001; MR1627486.
  • 19. Schick, Thomas, Real versus complex \(K\)-theory using Kasparov's bivariant \(KK\)-theory, Algebr. Geom. Topol., 4, 333-346 (2004); DOI 10.2140/agt.2004.4.333; zbl 1050.19003; MR2077669; arxiv math/0311295.
  • 20. Schick, Thomas, Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, 2014, Vol. II. The topology of scalar curvature, 1285-1307 (2014); zbl 1373.53053; MR3728662; arxiv 1405.4220.
  • 21. Wall, Terry; Scott, Peter, Topological methods in group theory. In: London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, Homological group theory (1977); zbl 0423.20023; MR0564422.
  • 22. tom Dieck, Tammo, Algebraic Topology, EMS Textbooks in Mathematics (2008), European Mathematical Society; DOI 10.4171/048; zbl 1156.55001; MR2456045.
  • 23. Valette, Alain, Introduction to the Baum-Connes conjecture, Lectures in Mathematics ETH Zürich (2002), Birkhäuser Verlag, Basel; DOI 10.1007/978-3-0348-8187-6; zbl 1136.58013; MR1907596.
  • 24. Egbert R. Van Kampen, On the connection between the fundamental groups of some related spaces, Amer. J. Math., 55, 1, 261-267 (1933); DOI 10.2307/2371128; zbl 59.0577.03; MR1506962.
  • 25. Zeidler, Rudolf, An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds, Algebr. Geom. Topol., 17, 5, 3081-3094 (2017); DOI 10.2140/agt.2017.17.3081; zbl 1377.58017; MR3704253; arxiv 1512.06781.

Affiliation

Nitsche, Martin
Institut für Geometrie, TU Dresden, Germany
Schick, Thomas
Mathematisches Institut, Universität Göttingen, Germany
Zeidler, Rudolf
Mathematisches Institut, WWU Münster, Germany

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