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Generalized Casson invariants for $ {\rm SO}(3),\;{\rm U}(2),\;{\rm Spin}(4),$ and $ {\rm SO}(4)$


Author: Cynthia L. Curtis
Journal: Trans. Amer. Math. Soc. 343 (1994), 49-86
MSC: Primary 57N10; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9947-1994-1207580-9
MathSciNet review: 1207580
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Abstract: We investigate Casson-type invariants corresponding to the low-rank groups $ {\text{SO}}(3)$, $ {\text{SU}}(2) \times {S^1}$, $ {\text{U}}(2)$, $ {\text{Spin}}(4)$ and $ {\text{SO}}(4)$. The invariants are defined following an approach similar to those of K. Walker and S. Cappell, R. Lee, and E. Miller. We obtain a description for each of the invariants in terms of the $ {\text{SU}}(2)$-invariant. Thus, all of them may be calculated using formulae for the $ {\text{SU}}(2)$-invariant. In defining these invariants, we offer methods which should prove useful for studying the invariants for other non-simply-connected groups once the invariants for the simply-connected covering groups are known.


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  • [A] M. F. Atiyah, New invariants of 3- and 4-dimensional manifolds, Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, R.I., 1988, pp. 285-299. MR 974342 (89m:57034)
  • [BL] S. Boyer and D. Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, UQAM Rapport de recherche 66, 1988.
  • [BN] S. Boyer and A. Nicas, Varieties of group representations and Casson's invariant for rational homology 3-spheres, Trans. Amer. Math. Soc. 322 (1990), 507-522. MR 972701 (92a:57020)
  • [C] A. Casson, Lectures at MSRI, 1985.
  • [CLM] S. Cappell, R. Lee, and E. Miller, A symplectic geometry approach to generalized Casson's invariants of 3-manifolds, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 269-275. MR 1016041 (91b:57024)
  • [G1] W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225. MR 762512 (86i:32042)
  • [G2] -, Representations of fundamental groups of surfaces, Geometry and Topology, Lecture Notes in Math., vol. 1167, Amer. Math. Soc., Providence, R.I., 1985, pp. 95-117. MR 827264 (87j:32068)
  • [Q] D. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Functional Anal. Appl. 19 (1985), 31-34. MR 783704 (86g:32035)
  • [W] K. Walker, An extension of Casson's invariant, Ann. of Math. Stud., no. 126, Princeton Univ. Press, Princeton, N. J., 1992. MR 1154798 (93e:57031)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1207580-9
Article copyright: © Copyright 1994 American Mathematical Society

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