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Twisted Donaldson invariants

Part of: Quantum field theory; related classical field theories Higher algebraic $K$-theory Infinite-dimensional manifolds

Published online by Cambridge University Press:  04 February 2021

TSUYOSHI KATO ,
HIROFUMI SASAHIRA  and
HANG WANG
TSUYOSHI KATO
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan. e-mail: [email protected]
HIROFUMI SASAHIRA
Affiliation:
Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan. e-mail: [email protected]
HANG WANG
Affiliation:
School of Mathematical Sciences, East China Normal University, South Lian Hua Road 5005 Minhang district, Shanghai200241, P. R. China. e-mail: [email protected]
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Abstract

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

MSC classification

Secondary: 58B34: Noncommutative geometry (à la Connes) 81T13: Yang-Mills and other gauge theories 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 515 - 568
Copyright
© Cambridge Philosophical Society 2021

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Footnotes

†

Supported by JSPS KAKENHI Grant Number JP17H02841

‡

Supported by JSPS KAKENHI Grant Number JP19K03493

§

Supported by grants NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200.

References

REFERENCES

Akbulut, S., Mrowka, T. and Ruan, Y.. Torsion classes and a universal constraint on Donaldson invariants for odd manifolds, Trans. Amer. Math. Soc. 347 (1995), 63–76.CrossRefGoogle Scholar
Atiyah, M. F., Singer, I. M.. The index of elliptic operators., IV. Ann. of Math. (2) 93 (1971), 119–138.CrossRefGoogle Scholar
Bauer, S. and Furuta, M.. A stable cohomotopy refinement of Seiberg–Witten invariants. I, Invent. Math. 155 (2004), 1–19.CrossRefGoogle Scholar
Blackdar, B.. K-theory for operator algebras. MSRI Publication Series 5 (Springer-Verlag, New York Heidelberg Berlin Tokyo, 1986).CrossRefGoogle Scholar
Burghelea, D.. The cyclic homology of the group rings. Comment. Math. Helv. 60 (1985), 354–365.CrossRefGoogle Scholar
Connes, A.. Noncommutative Gometry (Academic Press, Inc., San Diego, CA, 1994). xiv+661 pp.Google Scholar
Connes, A.. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360.CrossRefGoogle Scholar
Connes, A.. The action functional in non-commutative geometry. Comm. Math. Phys. 117 (1988), 673–683.CrossRefGoogle Scholar
Connes, A. and Moscovici, H.. Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29 (1990), no. 3, 345–388.CrossRefGoogle Scholar
Connes, A., Gromov, M. and Moscovoci, H.. Group cohomology with Lipschitz control and higher signatures. Geom. Funct. Anal. 3 (1993), no. 1, 1–78.CrossRefGoogle Scholar
Donaldson, S. K.. The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom. 26 (1987), no. 3, 397–428.CrossRefGoogle Scholar
Donaldson, S.K.. Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315.CrossRefGoogle Scholar
Donaldson, S. K.. Floer homology groups in Yang-Mills theory. With the assistance of Furuta, M. and Kotschick, D.. Cambridge Tracts in Mathematics, 147 (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
Donaldson, S. K. and Kronheimer, P. B.. The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1990), x+440 pp.Google Scholar
Emmanouil, I.. The Künneth formula in periodic cyclic homology, K-theory 10 (1996), 197–214.CrossRefGoogle Scholar
Friedman, R. and Morgan, J. W.. Smooth four-manifolds and complex surfaces. Ergeb. d. Mathe. Grenzgeb. 27 (Springer-Verlag, Berlin, 1994).Google Scholar
Gromov, M.. Hyperbolic groups. In Essays in group theory, Gersten, S.M., (Editor); MSRI Publications; 8 (Springer, New York, 1987), 75–264.CrossRefGoogle Scholar
Ishida, M. and Lebrun, C.. Curvature, connected sums and Seiberg–Witten theory, Comm. Anal. Geom. 11, (2003), 809–836.CrossRefGoogle Scholar
Ishida, M. and Sasahira, H.. Stable cohomotopy Seiberg-Witten invariants of connected sums of four-manifolds with positive first Betti number II: Applications, Comm. Anal. Geom. 25 (2017), no. 2, 373–393.CrossRefGoogle Scholar
Kasparov, G.G.. The operator K-functor and extensions of C*-algebras. (English Transl.). Math. USSR Izv. 16 (1981), 513–672.CrossRefGoogle Scholar
Kato, T.. Covering monopole map and higher degree in non commutative geometry, arXiv:1606.02402v1 (2016).Google Scholar
Kawauchi, A.. Splitting a 4-manifold with infinite cyclic fundamental group, Osaka J. Math. 31 (1994), no. 3, 489–495.Google Scholar
Kronheimer, P. B. and Mrowka, T. S.. Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 41 (1995), no. 3, 573–734.CrossRefGoogle Scholar
Lisca, P.. On the Donaldson polynomials of elliptic surfaces, Math. Ann. 299 (1994), no. 4, 629–639.CrossRefGoogle Scholar
Loday, J.-L.. Cyclic homology. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301. (Springer-Verlag, Berlin, 1998). xx+513 pp.CrossRefGoogle Scholar
Lott, J.. Superconnections and higher index theory. Geom. Funct. Anal. 2 (1992), no. 4, 421–454.CrossRefGoogle Scholar
Lott, J.. Diffeomorphisms and noncommutative analytic torsion., Mem. Amer. Math. Soc., 141, no. 673, (1999), viii+56 pp.Google Scholar
Lusztig, G.. Novikov’s higher signature and families of elliptic operators, J. Differential Geom. 7 (1971), 229–256.Google Scholar
Mishchenko, A.S.. Infinite dimensional representations of discrete groups and higher signatures, Izv. Akad. S.S.S. R. Ser. Mat. 38 (1974), 81–106.Google Scholar
Morgan, J. W. and Mrowka, T. S.. A note on Donaldson’s polynomial invariants, Internat. Math. Res. Notices (1992), no. 10, 223–230.CrossRefGoogle Scholar
Morgan, J. W. and Mrowka, T. S.. On the diffeomorphism classification of regular elliptic surfaces, Internat. Math. Res. Notices (1993), no. 6, 183–184.CrossRefGoogle Scholar
Morgan, J. W. and O’grady, K. G.. Differential topology of complex surfaces. Elliptic surfaces with pg = 1: smooth classification. With the collaboration of Millie Niss. Lecture Notes in Math., 1545 (Springer-Verlag, Berlin, 1993).Google Scholar
Sasahira, H.. Spin structures on Seiberg–Witten moduli spaces, J. Math. Sci. Univ. Tokyo. 13, 347–363 (2006).Google Scholar
Spanier, E. H.. Algebraic topology. Corrected reprint (Springer-Verlag, New York-Berlin, 1981).CrossRefGoogle Scholar