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arXiv:1104.2247v1[math.GT]12Apr2011 ACTION OF THE CORK TWIST ON FLOER HOMOLOGY
 

Summary: arXiv:1104.2247v1[math.GT]12Apr2011
ACTION OF THE CORK TWIST ON FLOER HOMOLOGY
SELMAN AKBULUT, C¸AGRI KARAKURT
Abstract. We utilize the Ozsv´ath-Szab´o contact invariant to detect the action of invo-
lutions on certain homology spheres that are surgeries on symmetric links, generalizing a
previous result of Akbulut and Durusoy. Potentially this may be useful to detect different
smooth structures on 4-manifolds by cork twisting operation.
1. Introduction
Any two different smooth structures of a closed simply connected 4-manifold are related
to each other by a cork twisting operation [16], and the cork can be assumed to be a Stein
manifold [4] (see [9] and [10] for applications). A quick way to generate corks, which was
used in [11], is from symmetric links as follows: Let L be a link in S3
with two components
K1 K2. Suppose that L satisfies the following:
(1) Both components K1 and K2 are unknotted.
(2) There is an involution of S3
exchanging K1 and K2.
(3) The linking number of K1 and K2 is ±1 (for some choice of orientations).
From this we can construct a 4­manifold W(L) by carving out a disk bounded by K1 from
4­ball, and attaching a 2­handle along K2 with framing 0. Therefore a handlebody diagram

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics