Walker, Kevin An extension of Casson’s invariant to rational homology spheres. (English) Zbl 0699.57008 Bull. Am. Math. Soc., New Ser. 22, No. 2, 261-267 (1990). Casson defined his invariant for oriented homology-3-spheres (integer coefficients). In the present note, describing results of the author’s thesis, a generalization of Casson’s construction is given extending the invariant to the case of rational homology-3-spheres (rational coefficients), again starting from representations of their fundamental groups in SU(2). A Dehn-surgery formula is given for the generalized invariant, which can be used to define it directly. For a further extension, see the next review (Zbl 0699.57009). Another generalization of Casson’s invariant to rational homology-3-spheres is given in a paper of Boyer and Nicas; see also a paper by S. Boyer and D. Lines [J. Reine Angew. Math. 405, 181-220 (1990; Zbl 0691.57004)], in which the case of homology lens spaces is considered. Reviewer: B.Zimmermann Cited in 7 ReviewsCited in 5 Documents MSC: 57N10 Topology of general \(3\)-manifolds (MSC2010) 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:representations of fundamental groups in SU(2); oriented homology-3- spheres; rational homology-3-spheres; Dehn-surgery; Casson’s invariant Citations:Zbl 0699.57009; Zbl 0691.57004 PDFBibTeX XMLCite \textit{K. Walker}, Bull. Am. Math. Soc., New Ser. 22, No. 2, 261--267 (1990; Zbl 0699.57008) Full Text: DOI References: [1] 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. · Zbl 0695.57011 [2] S. Boyer and D. Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, preprint. · Zbl 0691.57004 [3] B. I. Plotkin, Varieties of group representations, Uspehi Mat. Nauk 32 (1977), no. 5 (197), 3 – 68, 239 (Russian). [4] A. Casson, lectures at MSRI (1985). [5] William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200 – 225. · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 [6] C. McA. Gordon, Knots, homology spheres, and contractible 4-manifolds, Topology 14 (1975), 151 – 172. · Zbl 0304.57003 · doi:10.1016/0040-9383(75)90024-5 [7] F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Publish or Perish, Inc., Boston, Mass., 1974. Mathematics Lecture Series, No. 3. · Zbl 0288.10001 [8] Robion Kirby, A calculus for framed links in \?³, Invent. Math. 45 (1978), no. 1, 35 – 56. · Zbl 0377.55001 · doi:10.1007/BF01406222 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.