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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 2(188), Pages 69–134 (Mi umn3681)

Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis

A. S. Mishchenko

Abstract: This paper gives a survey of results on Hermitian $K$-theory over the last ten years. The main emphasis is on the computation of the numerical invariants of Hermitian forms with the help of the representation theory of discrete groups and by signature formulae on smooth multiply-connected manifolds.
In the first chapter we introduce the basic concepts of Hermitian $K$-theory. In particular, we discuss the periodicity property, the Bass-Novikov projections and new aids to the study of $K$-theory by means of representation spaces. In the second chapter we discuss the representation theory method for finding invariants of Hermitian forms. In § 5 we examine a new class of infinite-dimensional Fredholm representations of discrete groups. The third chapter is concerned with signature formulae on smooth manifolds and with various problems of differential topology in which the signature formulae find application.

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English version:
Russian Mathematical Surveys, 1976, 31:2, 71–138

Bibliographic databases:

UDC: 517.5+519.4
MSC: 19G38, 19D55, 57R20, 58B15

Citation: A. S. Mishchenko, “Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis”, Uspekhi Mat. Nauk, 31:2(188) (1976), 69–134; Russian Math. Surveys, 31:2 (1976), 71–138

Citation in format AMSBIB
\Bibitem{Mis76} \by A.~S.~Mishchenko \paper Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 2(188) \pages 69--134 \mathnet{http://mi.mathnet.ru/umn3681} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=413140} \zmath{https://zbmath.org/?q=an:0427.55001} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 2 \pages 71--138 \crossref{https://doi.org/10.1070/RM1976v031n02ABEH001478} 

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This publication is cited in the following articles:
1. A. S. Mishchenko, Yu. P. Solov'ev, “Representations of Banach algebras and formulas of Hirzebruch type”, Math. USSR-Sb., 39:2 (1981), 189–205
2. Anastasios Mallios, “Vector bundles and K-theory over topological algebras”, Journal of Mathematical Analysis and Applications, 92:2 (1983), 452
3. Anastasios Mallios, “Hermitian K-theory over topological ∗-algebras”, Journal of Mathematical Analysis and Applications, 106:2 (1985), 454
4. A. F. Kharshiladze, “Surgery on manifolds with finite fundamental groups”, Russian Math. Surveys, 42:4 (1987), 65–103
5. Shmuel Weinberger, “G-signatures and cyclotomic units”, Topology and its Applications, 32:2 (1989), 183
6. Maria H Papatriantafillou, “Differentiation in modules over topological ∗-algebras”, Journal of Mathematical Analysis and Applications, 170:1 (1992), 255
7. J. P. Levine, “Link invariants via the eta invariant”, Comment Math Helv, 69:1 (1994), 82
8. Donggeng Gong, Kefeng Liu, “Rigidity of higher elliptic genera”, Ann Global Anal Geom, 14:3 (1996), 219
9. A. A. Bolibrukh, A. A. Irmatov, M. I. Zelikin, O. B. Lupanov, V. M. Maynulov, E. F. Mishchenko, M. M. Postnikov, Yu. P. Solov'ev, E. V. Troitskii, “Aleksandr Sergeevich Mishchenko (on his 60th birthday)”, Russian Math. Surveys, 56:6 (2001), 1187–1191
10. Maria H. Papatriantafillou, “Homotopy classification of module bundles via Grassmannians”, Math Nachr, 280:1-2 (2007), 187
11. Kentaro Hori, Johannes Walcher, “D-brane categories for orientifolds—the Landau-Ginzburg case”, J High Energy Phys, 2008:4 (2008), 030
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