The Rochlin invariant of surgered, sewn link exteriors

Woodard, Mark R.

doi:10.1090/S0002-9939-1991-1034890-3

The Rochlin invariant of surgered, sewn link exteriors
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Proc. Amer. Math. Soc. 112 (1991), 211-221 Request permission

Abstract:

One method of producing $3$-manifolds is the sewing-up construction of W. R. Brakes. This involves identifying the boundary components of the exterior of a $2$-component link. Under certain assumptions, this process yields homology handles which can be surgered to obtain homology spheres. The main result is a formula for the Rochlin invariant of homology spheres obtained by this construction when the link is proper.

References

W. R. Brakes, Property $\textrm {R}$ and superslices, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 123, 263–281. MR 587091, DOI 10.1093/qmath/31.3.263
W. R. Brakes, Sewing-up link exteriors, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 27–37. MR 662425
Sylvain E. Cappell and Julius L. Shaneson, Some new four-manifolds, Ann. of Math. (2) 104 (1976), no. 1, 61–72. MR 418125, DOI 10.2307/1971056
James Eells Jr. and Nicolaas H. Kuiper, An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. (4) 60 (1962), 93–110. MR 156356, DOI 10.1007/BF02412768
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
F. Hirzebruch, W. D. Neumann, and S. S. Koh, Differentiable manifolds and quadratic forms, Lecture Notes in Pure and Applied Mathematics, Vol. 4, Marcel Dekker, Inc., New York, 1971. Appendix II by W. Scharlau. MR 0341499
Jim Hoste, Sewn-up $r$-link exteriors, Pacific J. Math. 112 (1984), no. 2, 347–382. MR 743990
Jim Hoste, The Arf invariant of a totally proper link, Topology Appl. 18 (1984), no. 2-3, 163–177. MR 769289, DOI 10.1016/0166-8641(84)90008-7
Akio Kawauchi, $H$-cobordism. I. The groups among three dimensional homology handles, Osaka Math. J. 13 (1976), no. 3, 567–590. MR 433474
Akio Kawauchi, On the Robertello invariants of proper links, Osaka J. Math. 21 (1984), no. 1, 81–90. MR 736972
K. Murasugi, On the Arf invariant of links, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 61–69. MR 727081, DOI 10.1017/S0305004100061314
Raymond A. Robertello, An invariant of knot cobordism, Comm. Pure Appl. Math. 18 (1965), 543–555. MR 182965, DOI 10.1002/cpa.3160180309
V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221–224 (Russian). MR 0052101
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288