Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic forms and the Birman-Craggs homomorphisms

Author: Dennis Johnson
Journal: Trans. Amer. Math. Soc. 261 (1980), 235-254
MSC: Primary 57N10; Secondary 57N05
MathSciNet review: 576873
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathcal{M}_g}$ be the mapping class group of a genus g orientable surface M, and $ {\mathcal{J}_g}$ the subgroup of those maps acting trivially on the homology group $ {H_1}(M,\,Z)$. Birman and Craggs produced homomorphisms from $ {\mathcal{J}_g}$ to $ {Z_2}$ via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on $ {H_1}(M,\,{Z_2})$ which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of $ {\mathcal{J}_g}$ by the intersection of the kernels of all these homomorphisms. It is a $ {Z_2}$-vector space isomorphic to a certain space of cubic polynomials over $ {H_1}(M,\,{Z_2})$. The dimension is then computed and found to be $ \left( {\begin{array}{*{20}{c}} {2g} \\ 3 \\ \end{array} } \right)\, + \,\left( {\begin{array}{*{20}{c}} {2g} \\ 2 \\ \end{array} } \right)$ . These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.

References [Enhancements On Off] (What's this?)

  • [A] C. Arf, Untersuchungen über Quadratische Formen in Körpern der Charakteristik 2, J. Reine Angew. Math. 183 (1941), 148-167. MR 0008069 (4:237f)
  • [BC] J. Birman and R. Craggs, The $ \mu $-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold, Trans. Amer. Math. Soc. 237 (1978), 283-309. MR 0482765 (58:2818)
  • [FK] Michael Freedman and Robion Kirby, A geometric proof of Rochlin's Theorem, Proc. Sympos. Pure Math., vol. 32, part 2, Amer. Math. Soc., Providence, R. I., 1978, pp. 85-97. MR 520525 (80f:57015)
  • [GA] F. Gonzalez-Acuña, Dehn's construction on knots, Bol. Soc. Mat. Méxicana 15 (1970), 58-79. MR 0356022 (50:8495)
  • [G] C. McA. Gordon, Knots, homology spheres, and contractible 4-manifolds, Topology 14 (1975), 151-172. MR 0402762 (53:6576)
  • [MH] Dale Husemoller and John Milnor, Symmetric bilinear forms, Ergebnisse der Math. und ihrer Grenzgebiete, vol. 73, Springer-Verlag, New York, 1973. MR 0506372 (58:22129)
  • [MS] F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, vol. 2, American Elsevier, New York, 1977. (See Chapter 13, Sections 2-5, particularly Theorems 4, 8, 12.)
  • [P] J. Powell, Two theorems on the mapping group of surfaces, Proc. Amer. Math. Soc. 68 (1978), 347-350. MR 0494115 (58:13045)
  • [W] F. Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology 7 (1968), 195-203. MR 0227992 (37:3576)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57N05

Retrieve articles in all journals with MSC: 57N10, 57N05

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society