Nonexistence of twists and surgeries generating exotic 4-manifolds

Yasui, Kouichi

doi:10.1090/tran/7696

Nonexistence of twists and surgeries generating exotic 4-manifolds
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Trans. Amer. Math. Soc. 372 (2019), 5375-5392 Request permission

Abstract:

It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer $n$, there exists a simply connected closed oriented 4-manifold $X$ such that for any compact (not necessarily connected) codimension zero submanifold $W$ with $b_1(\partial W)<n$, the set of all smooth structures on $X$ cannot be generated from $X$ by twisting $W$ and varying the gluing map. As a corollary, we show that there exists no “universal” compact 4-manifold $W$ such that for any simply connected closed 4-manifold $X$, the set of all smooth structures on $X$ is generated from a 4-manifold by twisting a fixed embedded copy of $W$ and varying the gluing map. Moreover, we give similar results for surgeries.

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