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arXiv:1303.2354math.GT

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

Ciprian Manolescu

Abstract

We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov'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 Rokhlin invariant one such that Y # 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.

arXiv comments
29 pages; final version, to appear in Journal of the AMS
Journal reference
DOI
Subjects
math.GTGeometric Topology math.ATAlgebraic Topology

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Discussed in comments2 threads

  1. Triangulation Conjecture Disproved 2 comments

    • Seems odd that they did not provide a counter example. My suspicion there is an incorrect assumption for non fractal structures. PS: This is really one of those cases where a paper does not mean much. You really need to dig through http://arxiv.org/abs/1303.2354 and find the…

      RetricView on HNParent context

    • Here is the actual paper: http://arxiv.org/pdf/1303.2354v4.pdf. It looks like this is their family of counter examples: A specific non-orientable five-dimensional manifold M5 with Sq1 ∆(M) ̸= 0 is constructed in [25]. Hence M5 is non-triangulable. To get non-orientable examples…

      panicView on HNParent context

  2. A Proof That Some Spaces Can’t Be Cut 1 comment

    • Seems that, though the posted article is new, the research in question is actually a couple of years old -- http://arxiv.org/abs/1303.2354

      pdpiView on HNParent context

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