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Trisections of surface complements and the Price twist

Seungwon Kim and Maggie Miller

Algebraic & Geometric Topology 20 (2020) 343–373

Given a real projective plane S embedded in a 4–manifold X4 with Euler number 2 or 2, the Price twist is a surgery operation on ν(S) yielding (up to) three different 4–manifolds: X4, τS(X4) and ΣS(X4). This is of particular interest when X4 = S4, as then ΣS(X4) is a homotopy 4–sphere which is not obviously diffeomorphic to S4. Here we show how to produce a trisection description of each Price twist on S X4 by producing a relative trisection of X4 ν(S). Moreover, we show how to produce a trisection description of general surface complements in 4–manifolds.

trisection, knotted surface, Price twist, surgery
Mathematical Subject Classification 2010
Primary: 57M50, 57R65
Received: 19 September 2018
Revised: 15 February 2019
Accepted: 16 May 2019
Published: 23 February 2020
Seungwon Kim
National Institute for Mathematical Sciences
South Korea
Center for Geometry and Physics
Institute for Basic Science (IBS)
Republic of Korea
Maggie Miller
Department of Mathematics
Princeton University
Princeton, NJ
United States