Varieties of group representations and Casson's invariant for rational homology -spheres
Authors:
S. Boyer and A. Nicas
Journal:
Trans. Amer. Math. Soc. 322 (1990), 507-522
MSC:
Primary 57N10
DOI:
https://doi.org/10.1090/S0002-9947-1990-0972701-6
MathSciNet review:
972701
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Andrew Casson's -valued invariant for
-homology
-spheres is shown to extend to a
-valued invariant for
-homology
-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian
and
representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian
or
representation is obtained. We also derive a sum theorem for Casson's invariant with respect to toroidal splittings of a
-homology
-sphere.
- [AM] Selman Akbulut and John D. McCarthy, Casson’s invariant for oriented homology 3-spheres, Mathematical Notes, vol. 36, Princeton University Press, Princeton, NJ, 1990. An exposition. MR 1030042
- [Ba] Hyman Bass, Finitely generated subgroups of 𝐺𝐿₂, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 127–136. MR 758465, https://doi.org/10.1016/S0079-8169(08)61638-4
- [B] Edward Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245–252. MR 0375356, https://doi.org/10.1016/0040-9383(75)90005-1
- [Br] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- [BZ] Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
- [FM]
S. Fukuhara and N. Maruyama, A sum formula for Casson's
-invariant, preprint.
- [Go] William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, https://doi.org/10.1016/0001-8708(84)90040-9
- [G1] C. McA. Gordon, Knots, homology spheres, and contractible 4-manifolds, Topology 14 (1975), 151–172. MR 0402762, https://doi.org/10.1016/0040-9383(75)90024-5
- [G2] C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708. MR 682725, https://doi.org/10.1090/S0002-9947-1983-0682725-0
- [HW] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, 1960.
- [Hi] Heisuke Hironaka, Triangulations of algebraic sets, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185. MR 0374131
- [Ja] William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
- [JS] William H. Jaco and Peter B. Shalen, Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, https://doi.org/10.1090/memo/0220
- [Jo] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
- [Mi] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- [Mo] Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 0383406
- [Mu] David Mumford, Algebraic geometry. I, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR 0453732
- [N] Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125–144. MR 585657
- [Oe] Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067
- [Sh] Peter Shalen, The proof in the case of no incompressible surface, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 21–36. MR 758463, https://doi.org/10.1016/S0079-8169(08)61636-0
- [T]
W. Thurston, The geometry and topology of
-manifolds, Princeton Univ. Lecture Notes, 1977.
- [W] André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 0169956, https://doi.org/10.2307/1970495
- [Wo] Joseph A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Inc., Boston, Mass., 1974. MR 0343214
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1990-0972701-6
Keywords:
Casson's invariant,
representations,
Zariski tangent space
Article copyright:
© Copyright 1990
American Mathematical Society