Generalized Casson invariants for and

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate Casson-type invariants corresponding to the low-rank groups , , , and . 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 -invariant. Thus, all of them may be calculated using formulae for the -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.

**[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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57N10,
57M05

Retrieve articles in all journals with MSC: 57N10, 57M05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1207580-9

Article copyright:
© Copyright 1994
American Mathematical Society