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 Izv. Akad. Nauk SSSR Ser. Mat., 1970, Volume 34, Issue 2, Pages 253–288 (Mi izv2415)

Algebraic construction and properties of hermitian analogs of $K$-theory over rings with involution from the viewpoint of hamiltonian formalism. applications to differential topology and the theory of characteristic classes. I

S. P. Novikov

Abstract: The complicated and intricate algebraic material in smooth topology (the theory of surgery) does not fit into the already existing concepts of stable algebra. It turns out that the systematization of this material is most naturally carried through from the point of view of an algebraic version of the hamiltonian formalism over rings with involution. The present article is devoted to this task. The first part contains a development of the algebraic concepts.

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English version:
Mathematics of the USSR-Izvestiya, 1970, 4:2, 257–292

Bibliographic databases:

UDC: 513.8
MSC: 19G38, 57R20, 16W10, 55N20, 18F25

Citation: S. P. Novikov, “Algebraic construction and properties of hermitian analogs of $K$-theory over rings with involution from the viewpoint of hamiltonian formalism. applications to differential topology and the theory of characteristic classes. I”, Izv. Akad. Nauk SSSR Ser. Mat., 34:2 (1970), 253–288; Math. USSR-Izv., 4:2 (1970), 257–292

Citation in format AMSBIB
\Bibitem{Nov70} \by S.~P.~Novikov \paper Algebraic construction and properties of hermitian analogs of $K$-theory over rings with involution from the viewpoint of hamiltonian formalism. applications to differential topology and the theory of characteristic classes.~I \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1970 \vol 34 \issue 2 \pages 253--288 \mathnet{http://mi.mathnet.ru/izv2415} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=292913} \zmath{https://zbmath.org/?q=an:0193.51902|0216.45003} \transl \jour Math. USSR-Izv. \yr 1970 \vol 4 \issue 2 \pages 257--292 \crossref{https://doi.org/10.1070/IM1970v004n02ABEH000903} 

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This publication is cited in the following articles:
1. S. P. Novikov, “Algebraic construction and properties of Hermitian analogs of $K$-theory over rings with involution from the viewpoint of Hamiltonian formalism. applications to differential topology and the theory of characteristic classes. II”, Math. USSR-Izv., 4:3 (1970), 479–505
2. I. A. Volodin, “Algebraic $K$-theory as extraordinary homology theory on the category of associative rings with unity”, Math. USSR-Izv., 5:4 (1971), 859–887
3. A. S. Mishchenko, “Infinite-dimensional representations of discrete groups, and higher signatures”, Math. USSR-Izv., 8:1 (1974), 85–111
4. V. P. Maslov, M. V. Fedoryuk, “The canonic operator (real case)”, J Math Sci, 3:2 (1975), 217
5. A. F. Kharshiladze, “Manifolds of the homotopy type of the product of two projective spaces”, Math. USSR-Sb., 25:4 (1975), 471–486
6. A. S. Mishchenko, “Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis”, Russian Math. Surveys, 31:2 (1976), 71–138
7. L. N. Vaserstein, “Foundations of algebraic $K$-theory”, Russian Math. Surveys, 31:4 (1976), 89–156
8. V. A. Vardanyan, “On periodicity in Hermitian theories”, Russian Math. Surveys, 33:2 (1978), 253–254
9. V. E. Nazaikinskii, V. G. Oshmyan, B. Yu. Sternin, V. E. Shatalov, “Fourier integral operators and the canonical operator”, Russian Math. Surveys, 36:2 (1981), 93–161
10. V. G. Turaev, “A cocycle for the symplectic first chern class and the maslov index”, Funct. Anal. Appl., 18:1 (1984), 35–39
11. A. F. Kharshiladze, “Surgery on manifolds with finite fundamental groups”, Russian Math. Surveys, 42:4 (1987), 65–103
12. S. P. Novikov, “The Schrödinger operator on graphs and topology”, Russian Math. Surveys, 52:6 (1997), 1320–1321
13. Yu. V. Muranov, I. Hambleton, “Projective splitting obstruction groups for one-sided submanifolds”, Sb. Math., 190:10 (1999), 1465–1485
14. Cavicchioli A. Muranov Y. Repovs D., “Algebraic Properties of Decorated Splitting Obstruction Groups”, Boll. Unione Mat. Italiana, 4B:3 (2001), 647–675
15. A. Ranicki, “Blanchfield and Seifert algebra in high-dimensional knot theory”, Mosc. Math. J., 3:4 (2003), 1333–1367
16. Nigel Higson, John Roe, “Mapping Surgery to Analysis I: Analytic Signatures”, K-Theory, 33:4 (2004), 277
17. Cavicchioli A. Muranov Y. Spaggiari F., “Relative Groups in Surgery Theory”, Bull. Belg. Math. Soc.-Simon Steven, 12:1 (2005), 109–135
18. St. Petersburg Math. J., 19:1 (2008), 159–165
19. A. Bak, Yu. V. Muranov, “Splitting a simple homotopy equivalence along a submanifold with filtration”, Sb. Math., 199:6 (2008), 787–809
20. EDGAR G. GOODAIRE, CÉSAR POLCINO MILIES, “ORIENTED INVOLUTIONS AND SKEW-SYMMETRIC ELEMENTS IN GROUP RINGS”, J. Algebra Appl, 12:01 (2013), 1250131
21. P. G. Grinevich, S. P. Novikov, “Discrete $SL_n$-connections and self-adjoint difference operators on two-dimensional manifolds”, Russian Math. Surveys, 68:5 (2013), 861–887
22. G. Yu, “The Novikov conjecture”, Russian Math. Surveys, 74:3 (2019), 525–541
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