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Degree‐one maps, surgery and four‐manifolds
Author(s) -
Gadgil Siddhartha
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm019
Subject(s) - mathematics , unknot , degree (music) , embedding , manifold (fluid mechanics) , pure mathematics , boundary (topology) , combinatorics , topology (electrical circuits) , mathematical analysis , knot (papermaking) , computer science , physics , acoustics , mechanical engineering , chemical engineering , artificial intelligence , engineering
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 × I ) # nCP 2¯ # mCP 2, for some m ⩾ 0, n ⩾ 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.