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HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

  • MATTHIAS KRECK (a1) and HAGGAI TENE (a2)

Abstract

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$ , which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$ .

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[1] Atiyah, M. F., K-theory, Lecture notes by D. W. Anderson (W. A. Benjamin, Inc., New York–Amsterdam, 1967).
[2] Baker, A. and Özel, C., ‘Complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups’, in(English Summary) Geometry and Topology: Aarhus (1998), Contemporary Mathematics, 258 (American Mathematical Society, Providence, RI, 2000), 119.
[3] Burghelea, D. and Henderson, D., ‘Smoothings and homeomorphisms for Hilbert manifolds’, Bull. Amer. Math. Soc. (N.S.) 76 (1970).
[4] Burghelea, D. and Kuiper, N. H., ‘Hilbert manifolds’, Ann. of Math. (2) 90(3) (1969), 379417.
[5] Conner, P. E. and Floyd, E. E., Differentiable Periodic Maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 33 (Academic Press Inc., Springer, Berlin–Göttingen–Heidelberg, New York, 1964).
[6] Dold, A., Geometric Cobordism and the Fixed Point Transfer, Lecture Notes in Mathematics (Springer, Berlin, 1976).
[7] Eells, J., Fibring Spaces of Maps, Symposium on Infinite-Dimensional Topology (Louisiana State University, Baton Rouge, LA, 1967), Annals of Mathematics Studies, 69 (Princeton University Press, Princeton, NJ, 1972), 4357.
[8] Eells, J., ‘A setting for global analysis’, Bull. Amer. Math. Soc. (N.S.) 72 (1966), 751807.
[9] Eells, J. and Elworthy, K. D., ‘Open embeddings of certain Banach manifolds’, Ann. of Math. (2) 91(3) (1970).
[10] Elworthy, K. D. and Tromba, A. J., ‘Degree theory on Banach manifolds’, inNonlinear Functional Analysis, Proceedings of Symposia in Pure Mathematics, Vol. XVIII, Part 1, Chicago, IL, 1968 (American Mathematical Society, Providence, RI, 1970), 8694.
[11] Geba, K. and Granas, A., ‘Infinite dimensional cohomology theories’, J. Math. Pures Appl. (9) 52 (1973).
[12] Goresky, M. and Hingston, N., ‘Loop products and closed geodesics’, Duke Math. J. 150(1) (2009), 117209.
[13] Anna, G., Resolution of stratifolds and connection to Mather’s abstract pre-stratified spaces, PhD Heidelberg, 2002, http://archiv.ub.uni-heidelberg.de/volltextserver/3127/1/grinberg_diss.pdf.
[14] Henderson, D. W., ‘Infinite-dimensional manifolds are open subsets of Hilbert space’, Bull. Amer. Math. Soc. (N.S.) 75(4) (1969), 759762.
[15] Hirsch, M. W., Differential Topology, Graduate Texts in Mathematics, 33 (Springer, New York–Heidelberg, 1976).
[16] Jänich, K., ‘Vektorraumbündel und der Raum der Fredholm-Operatoren’, Math. Ann. 161 (1965).
[17] Koschorke, U., ‘Infinite dimensional K-theory and characteristic classes of Fredholm bundle maps’, PhD Thesis, Brandeis University, 1968.
[18] Kreck, M., Differential Algebraic Topology: From Stratifolds to Exotic Spheres, Graduate Studies in Mathematics, 110 (American Mathematical Society, 2010).
[19] Kreck, M. and Singhof, W., ‘Homology and cohomology theories on manifolds’, Münster J. Math. 3 (2010), 19.
[20] Kreck, M. and Tene, H., Extending functors on the category of manifolds and submersions, Manifold Atlas. http://www.map.mpim-bonn.mpg.de/Extending_functors_on_the_category_of_manifolds_and_submersions.
[21] Kreck, M. and Tene, H., Orientation of Fredholm maps, Manifold Atlas. http://www.map.mpim-bonn.mpg.de/Orientation_of_Fredholm_maps_between_Hilbert_manifolds.
[22] Kuiper, N. H., ‘The homotopy type of the unitary group of Hilbert space’, Topology 3(1) (1965).
[23] Lang, S., Fundamentals of Differential Geometry, Graduate Texts in Mathematics, 191 (Springer, New York, 1999).
[24] MacPherson, R., Equivariant Invariants and Linear Geometry, Geometric Combinatorics, IAS/Park City Mathematics Series, 13 (American Mathematical Society (AMS), Providence, RI, 2007, Princeton, NJ: Institute for Advanced Studies), 317–388.
[25] Mather, J., ‘Notes on topological stability’, Bull. Amer. Math. Soc. (N.S.) 49(4) (2012), 475506.
[26] Milnor, J. W., ‘On axiomatic homology theory’, Pacific J. Math. 12(1) (1962).
[27] Morava, J. J., ‘Algebraic topology of Fredholm maps’, Doctoral Thesis, Rice university, 1968, http://hdl.handle.net/1911/19013.
[28] Mukherjea, K. K., ‘Cohomology theory for Banach manifolds’, J. Math. Mech. 19 (1969/1970), 731744.
[29] Palais, R. S., ‘Morse theory on Hilbert manifolds’, Topology 2 (1963), 299340.
[30] Palais, R. S., ‘When proper maps are closed’, Proc. Amer. Math. Soc. 24 (1970), 835836.
[31] Pressley, A. and Segal, G., Loop Groups, Oxford Mathematical Monographs, Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1986).
[32] Quillen, D., ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math. 7(1) (1971), 2956.
[33] Quillen, D., ‘Determinants of Cauchy–Riemann operators over a Riemann surface’, inFunctional Analysis and Its Applications 19 (Springer, New York, 1985), 3134.
[34] Smale, S., ‘An infinite dimensional version of Sard’s theorem’, Am. J. Math. 87(4) (1965).
[35] Tene, H., ‘Stratifolds and equivariant cohomology theories’, PhD Thesis, University of Bonn, 2010, http://hss.ulb.uni-bonn.de/2011/2391/2391.pdf.
[36] Tene, H., Some geometric equivariant cohomology theories, http://arxiv.org/pdf/1210.7923v2.pdf.
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HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

  • MATTHIAS KRECK (a1) and HAGGAI TENE (a2)

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