# The Conley index, gauge theory, and triangulations

@article{Manolescu2013TheCI,
title={The Conley index, gauge theory, and triangulations},
author={Ciprian Manolescu},
journal={Journal of Fixed Point Theory and Applications},
year={2013},
volume={13},
pages={431-457}
}
• Ciprian Manolescu
• Published 29 August 2013
• Mathematics
• Journal of Fixed Point Theory and Applications
This is an expository paper about Seiberg–Witten Floer stable homotopy types.We outline their construction, which is based on the Conley index and finite-dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.
10 Citations

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## References

SHOWING 1-10 OF 99 REFERENCES
CONNECTED SIMPLE SYSTEMS AND THE CONLEY INDEX OF ISOLATED INVARIANT SETS
The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central
A stable cohomotopy refinement of Seiberg-Witten invariants: II
A gluing theorem for the stable cohomotopy invariant defined in the first article in this series of two gives new results on diffeomorphism types of decomposable manifolds.
Equivariant Seiberg-Witten Floer Homology
• Mathematics
• 1996
This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around
Monopoles and Three-Manifolds
• Mathematics
• 2008
Preface 1. Outlines 2. The Seiberg-Witten equations and compactness 3. Hilbert manifolds and perturbations 4. Moduli spaces and transversality 5. Compactness and gluing 6. Floer homology 7.
A stable cohomotopy refinement of Seiberg-Witten invariants: I
• Mathematics
• 2002
The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. Part I discusses the definition of this stable homotopy invariant and its relation
Aspherical manifolds that cannot be triangulated
• Mathematics
• 2014
By a result of Manolescu there are topological closed n-manifolds that cannot be triangulated for each n 5. We show here that for n 6 we can choose such manifolds to be aspherical. AMS classication
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 periodic
WITTEN'S COMPLEX AND INFINITE DIMENSIONAL MORSE THEORY
We investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. The results are applied to an
Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture
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
Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0
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