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Disoriented homology and double branched covers

Part of: Low-dimensional topology

Published online by Cambridge University Press:  11 November 2022

Brendan Owens Open the ORCID record for Brendan Owens [Opens in a new window]  and
Sašo Strle
Brendan Owens*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom
Sašo Strle
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia e-mail: saso.strle@fmf.uni-lj.si
*
e-mail: brendan.owens@glasgow.ac.uk
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Abstract

This paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.

To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.

Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

Keywords

Disoriented homologyintersection formdouble branched cover

MSC classification

Primary: 57M12: Special coverings, e.g. branched
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 43
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The second author was partially supported by Slovenian Research Agency (ARRS) Research program P1-0288.

References

Akbulut, S., 4-manifolds, Oxford Graduate Texts in Mathematics, 25, Oxford University Press, Oxford, 2016.CrossRefGoogle Scholar
Akbulut, S. and Kirby, R., Branched covers of surfaces in $4$ -manifolds . Math. Ann. 252(1979/80), no. 2, 111–131.CrossRefGoogle Scholar
Carter, S., Kamada, S., and Saito, M., Surfaces in 4-space, Encyclopaedia of Mathematical Sciences, 142, Springer, Berlin, 2004, Low-dimensional topology, III.Google Scholar
Gompf, R. E. and Stipsicz, A. I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20, American Mathematical Society, Providence, RI, 1999.Google Scholar
Gordon, C. M. A. and Litherland, R. A., On the signature of a link . Invent. Math. 47(1978), no. 1, 53–69.CrossRefGoogle Scholar
Greene, J. E., Lattices, graphs, and Conway mutation . Invent. Math. 192(2013), no. 3, 717–750.CrossRefGoogle Scholar
Greene, J. E., A spanning tree model for the Heegaard Floer homology of a branched double-cover . J. Topol. 6(2013), no. 2, 525–567.CrossRefGoogle Scholar
Grigsby, J. E., Knot Floer homology in cyclic branched covers . Algebr. Geom. Topol. 6(2006), 1355–1398.Google Scholar
Ishiwata, M., Przytycki, J. H., and Yasuhara, A., Branched covers of tangles in three-balls . Canad. Math. Bull. 46(2003), no. 3, 356–364.CrossRefGoogle Scholar
Lisca, P., Lens spaces, rational balls and the ribbon conjecture . Geom. Topol. 11(2007), 429–472.Google Scholar
Livingston, C., Knot theory, Carus Mathematical Monographs, 24, Mathematical Association of America, Washington, DC, 1993.CrossRefGoogle Scholar
Manolescu, C., Nilpotent slices, Hilbert schemes, and the Jones polynomial . Duke Math. J. 132(2006), no. 2, 311–369.Google Scholar
Owens, B., Knots and 4-manifolds . Winter Braids Lect. Notes 6(2019), no. IX, Exp. no. II, 26 pp.Google Scholar
Ozsváth, P. and Szabó, Z., On the Heegaard Floer homology of branched double-covers . Adv. Math. 194(2005), no. 1, 1–33.CrossRefGoogle Scholar
Rolfsen, D., Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990, Corrected reprint of the 1976 original.Google Scholar