A non-abelian Seiberg-Witten invariant for integral homology 3-spheres (original) (raw)

Seiberg-Witten invariant and Casson-Walker invariant for rational homology 3-spheres

We consider a modi ed version of the Seiberg{Witten invariants for rational homology 3{spheres, obtained by adding to the original invariants a correction term which is a combination of {invariants. We show that these modi ed invariants are topological invariants. We prove that an averaged version of these modi ed invariants equals the Casson{Walker invariant. In particular, this result proves an averaged version of a conjecture of Ozsv ath and Szab o on the equivalence between their ̂ invariant and the Seiberg{Witten invariant of rational homology 3{spheres.

Seiberg–Witten and Casson–Walker Invariants for Rational Homology 3-Spheres

2002

We consider a modified version of the Seiberg-Witten invariants for rational homology 3-spheres, obtained by adding to the original invariants a correction term which is a combination of Z-invariants. We show that these modified invariants are topological invariants. We prove that an averaged version of these modified invariants equals the Casson-Walker invariant. In particular, this result proves an averaged version of a conjecture of Ozsva´th and Szabo´on the equivalence between theirŷ invariant and the Seiberg-Witten invariant of rational homology 3-spheres.

6 Seiberg-Witten-Floer Theory for Homology 3-Spheres

1996

We give the definition of the Seiberg-Witten-Floer homology group for a homology 3-sphere. Its Euler characteristic number is a Casson-type invariant. For a four-manifold with boundary a homology sphere, a relative Seiberg-Witten invariant is defined taking values in the Seiberg-Witten-Floer homology group, these relative Seiberg-Witten invariants are applied to certain homology spheres bounding Stein surfaces.

Defining an SU(3)-Casson/U(2)-Seiberg-Witten integer invariant for integral homology 3-spheres

2004

The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat SU(3)-connections. This obstruction is the non-trivial spectral flow of a family of twisted signature operators in 3-dimensions. The parallel U(2)-Seiberg-Witten construction also has an obstruction but from the non-trivial spectral flow of a family of twisted Dirac operators. By taking the SU(3)-flat and U(2)-Seiberg-Witten equations simultaneously the obstructions can be made to cancel and an integer invariant is obtained.

Seiberg-Witten-Floer Theory for Homology 3-Spheres

Seiberg-Witten Invariants for 3-manifolds

2015

In this note we give a detailed exposition of the Seiberg-Witten invariants for closed oriented 3-manifolds paying par-ticular attention to the case of b1 = 0 and b1 = 1. These are extracted from the moduli space of solutions to the Seiberg-Witten equations which depend on choices of a Riemannian metric on the underlying manifold as well as certain pertur-bation terms in the equations. In favourable circumstances this moduli space is finite and naturally oriented and we may form the algebraic sum of the points. Given any two sets of choices of metric and perturbation which are connected by a 1-parameter family, we analyse in detail the singular-ities which may develop in the interpolating moduli space. This leads then to an understanding of how the algebraic sum changes. In the case b1 = 0 a topological invariant can be extracted with the addition of a suitable counter-term, which we identify (this idea is attributed to Donaldson). In the case b1 = 1 a topological invariant is defin...

Unified SO(3) Quantum Invariants for Rational Homology 3-Spheres

arXiv (Cornell University), 2008

Given a rational homology 3-sphere M with |H 1 (M, Z)| = b and a link L inside M , colored by odd numbers, we construct a unified invariant I M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M, L). If b = 1 and L = ∅, I M coincides with Habiro's invariant of integral homology 3-spheres. For b > 1, the unified invariant defined by the third author is determined by I M. One of the applications are the new Ohtsuki series (perturbative expansions of I M at roots of unity) dominating all quantum SO(3) invariants.

A non-abelian Seiberg-Witten invariant for integral homology 3-spheres (original) (raw)

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