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

@article{Manolescu2013Pin2equivariantSF,
title={Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture},
author={Ciprian Manolescu},
journal={arXiv: Geometric Topology},
year={2013}
}
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… Expand
100 Citations

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#### References

SHOWING 1-10 OF 85 REFERENCES
Seiberg{Witten{Floer stable homotopy type of three-manifolds with b1 =0
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in theExpand
W-invariants and Neumann–Siebenmann invariants for Seifert homology 3-spheres
• Mathematics
• 2001
Abstract We give a reciprocity formula for w-invariants for homology 3-spheres defined by Fukumoto and Furuta, which enables us to compute these invariants recursively. Using this formula, we showExpand
Instanton Homology of Seifert Fibred Homology Three Spheres
• Mathematics
• 1990
For an oriented integral homology 3-sphere 2, A. Casson has introduced an integer invariant A(2) that is defined by using the space £%(2) of conjugacy classes of irreducible representations of ^ ( 2Expand
Equivariant aspects of Yang–Mills Floer theory
Abstract We study the u-map in instanton Floer homology using Floer's exact surgery triangle. As an application we prove that the Donaldson invariants of simply-connected smooth 4-manifolds haveExpand
PERIODIC FLOER PRO-SPECTRA FROM THE SEIBERG-WITTEN EQUATIONS
• Mathematics
• 2003
Given a three-manifold with b1 = 1 and a nontorsion spin c structure, we use finite dimensional approximation to construct from the Seiberg-Witten equations two invariants in the form of a periodicExpand
FLOER HOMOLOGY AND INVARIANTS OF HOMOLOGY COBORDISM
By using surgery techniques, we compute Floer homology for certain classes of integral homology 3-spheres homology cobordant to zero. We prove that Floer homology is two-periodic for all theseExpand
Rohlin's invariant and gauge theory III. Homology 4–tori.
• Mathematics
• 2005
This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theo- rem. Such relations are well-known inExpand
A concordance invariant from the Floer homology of double branched covers
• Mathematics
• 2005
Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branchedExpand
A gluing theorem for the relative Bauer-Furuta invariants
In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of theExpand
Fukumoto-Furuta invariants of plumbed homology 3-spheres
Recent major progress in the study of the homology cobordism group of homology 3-spheres is related to the work of Fukumoto, Furuta, and Ue on their w-invariant. We identify the w-invariant with theExpand