Varieties of group representations and Casson’s invariant for rational homology $3$-spheres
HTML articles powered by AMS MathViewer
- by S. Boyer and A. Nicas PDF
- Trans. Amer. Math. Soc. 322 (1990), 507-522 Request permission
Abstract:
Andrew Casson’s ${\mathbf {Z}}$-valued invariant for ${\mathbf {Z}}$-homology $3$-spheres is shown to extend to a ${\mathbf {Q}}$-valued invariant for ${\mathbf {Q}}$-homology $3$-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian ${\operatorname {SL} _2}({\mathbf {C}})$ and $\operatorname {SU} (2)$ representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian ${\operatorname {SL} _2}({\mathbf {C}})$ or $\operatorname {SU} (2)$ representation is obtained. We also derive a sum theorem for Casson’s invariant with respect to toroidal splittings of a ${\mathbf {Z}}$-homology $3$-sphere.References
- 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, DOI 10.1515/9781400860623
- Hyman Bass, Finitely generated subgroups of $\textrm {GL}_{2}$, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 127–136. MR 758465, DOI 10.1016/S0079-8169(08)61638-4
- Edward Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245–252. MR 375356, DOI 10.1016/0040-9383(75)90005-1
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
- Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776 S. Fukuhara and N. Maruyama, A sum formula for Casson’s $\lambda$-invariant, preprint.
- William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, DOI 10.1016/0001-8708(84)90040-9
- C. McA. Gordon, Knots, homology spheres, and contractible $4$-manifolds, Topology 14 (1975), 151–172. MR 402762, DOI 10.1016/0040-9383(75)90024-5
- C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708. MR 682725, DOI 10.1090/S0002-9947-1983-0682725-0 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, 1960.
- 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
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
- 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, DOI 10.1090/memo/0220
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- 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
- Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406, DOI 10.2140/pjm.1971.38.737
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- 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
- Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067, DOI 10.2140/pjm.1984.111.209
- 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, DOI 10.1016/S0079-8169(08)61636-0 W. Thurston, The geometry and topology of $3$-manifolds, Princeton Univ. Lecture Notes, 1977.
- André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 169956, DOI 10.2307/1970495
- Joseph A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Inc., Boston, Mass., 1974. MR 0343214
Additional Information
- © Copyright 1990 American Mathematical Society
- 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