Degree-one maps, surgery and four-manifolds

Gadgil, Siddhartha (2007) Degree-one maps, surgery and four-manifolds. In: Bulletin of the London Mathematical Society, 39 (3). pp. 419-424.

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Abstract

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold M to a closed, oriented 3-manifold N if and only if M can be obtained from N by surgery about a link in N each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from M to N if and only if there is a smooth embedding of M in $W = (N \times I)\#_n\bar{CP^2}\#_mCP^2$, for some $m\ge 0$, $n\ge 0$ which separates the boundary components of W. This is motivated by the relation to topological field theories, in particular the invariants of Ozsvath and Szabo.

Item Type: Journal Article Copyright of this article belongs to London Mathematical Society. Division of Physical & Mathematical Sciences > Mathematics 11 Jun 2008 19 Sep 2010 04:45 http://eprints.iisc.ernet.in/id/eprint/14285