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Pin(2)-Equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture


Author: Ciprian Manolescu
Journal: J. Amer. Math. Soc. 29 (2016), 147-176
MSC (2010): Primary 57R58; Secondary 57Q15, 57M27
DOI: https://doi.org/10.1090/jams829
Published electronically: April 22, 2015
MathSciNet review: 3402697
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Abstract: We define $ \operatorname {Pin}(2)$-equivariant Seiberg-Witten Floer homology for rational homology $ 3$-spheres equipped with a spin structure. The analogue of Frøyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod $ 2$ reduction is the Rokhlin invariant. As an application, we show that there are no homology $ 3$-spheres $ Y$ of the Rokhlin invariant one such that $ Y \char93 Y$ bounds an acyclic smooth $ 4$-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.


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Additional Information

Ciprian Manolescu
Affiliation: Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, California 90095
Email: cm@math.ucla.edu

DOI: https://doi.org/10.1090/jams829
Received by editor(s): April 10, 2013
Received by editor(s) in revised form: February 6, 2014, September 17, 2014, and October 6, 2014
Published electronically: April 22, 2015
Additional Notes: The author was supported by NSF grant DMS-1104406.
Article copyright: © Copyright 2015 American Mathematical Society

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