Characteristic cohomotopy classes for families of 4-manifolds

Szymik, Markus

doi:10.1515/forum.2010.027

Cited by 20 publications

(24 citation statements)

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“…We illustrate now an example where our constructions in terms of perturbation of Fredholm morphisms appear in a natural way. This approach was used by Stefan Bauer [3] and Mikio Furuta, see also [19] and [26].…”

Section: Example 42 (G-euclidean Neighborhood Retractsmentioning

confidence: 99%

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Nonlinearity, proper actions and equivariant stable cohomotopy

Torres

2013

Pure and Applied Mathematics Quarterly

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Section: Example 42 (G-euclidean Neighborhood Retractsmentioning

confidence: 99%

“…The crucial result in this is the construction of an index theory in the context of parametrized nonlinear analysis. Previous versions of this index theory (restricted to the unparametrized and S 1equivariant case) were constructed in [3], [26], [19] from where we adopt the crucial ideas.…”

Section: L+(log(t)−(log(−t)))(c(p(v))mentioning

confidence: 99%

Nonlinearity, proper actions and equivariant stable cohomotopy

Torres

2013

Pure and Applied Mathematics Quarterly

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“…The aim of this section is to show that the Bauer-Furuta invariants, see [BF04], can be used to define stable characteristic classes for families of 4-manifolds. Let me first recall how they have been extended, in [Szy10], to define unstable characteristic classes for families of 4-manifolds. From now on, as already in the previous section, all 4-manifolds will be assumed to be simply-connected.…”

Section: Bauer-furuta Invariantsmentioning

confidence: 99%

Stable Diffeomorphism Groups of~4-Manifolds

Szymik

2008

Mathematical Research Letters

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Abstract. A localisation of the category of n-manifolds is introduced by formally inverting the connected sum construction with a chosen n-manifold Y . On the level of automorphism groups, this leads to the stable diffeomorphism groups of n-manifolds. In dimensions 0 and 2, this is connected to the stable homotopy groups of spheres and the stable mapping class groups of Riemann surfaces. In dimension 4 there are many essentially different candidates for the n-manifold Y to choose from. It is shown that the Bauer-Furuta invariants provide invariants in the case Y = CP 2 , which is related to the birational classification of complex surfaces. This will be the case for other Y only after localisation of the target category. In this context, it is shown that the K3-stable Bauer-Furuta invariants determine the S 2 ×S 2 -stable invariants.

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“…Naturally one may also ask for a families extension of the Bauer-Furuta invariant. The existence of such an extension can already been seen implicitly in Theorem 2.6]) and was further developed by Szymik in [26]. Neither of these works establish how, if at all, one can recover the families Seiberg-Witten invariants from the families Bauer-Furuta invariant.…”

Section: Introductionmentioning

confidence: 99%

“…In [3] Bauer and Furuta constructed a stable cohomotopy refinement of the Seiberg-Witten invariant by taking a finite dimensional approximation of the Seiberg-Witten equations. This construction was extended to families of 4-manifolds in [26]. We will recall how this finite dimensional approximation is constructed in Subsection 2.3.…”

Section: Introductionmentioning

confidence: 99%

On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds

Baraglia,

Konno

2019

Preprint

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We show how the families Seiberg-Witten invariants of a family of smooth 4-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the families Seiberg-Witten invariants leading to a series of mod 2 relations between these invariants and the Chern classes of the spin c index bundle of the family. As a result we discover a new aspect of the ordinary Seiberg-Witten invariants of a 4-manifold X: they obstruct the existence of certain families of 4-manifolds with fibres diffeomorphic to X. As a concrete geometric application, we shall detect a non-smoothable family of K3 surfaces. Our formalism also leads to a simple new proof of the families wall crossing formula. Lastly, we introduce K-theoretic Seiberg-Witten invariants and give a formula expressing the Chern character of the K-theoretic Seiberg-Witten invariants in terms of the cohomological Seiberg-Witten invariants. This leads to new divisibility properties of the families Seiberg-Witten invariants.

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Characteristic cohomotopy classes for families of 4-manifolds

Cited by 20 publications

References 16 publications

Nonlinearity, proper actions and equivariant stable cohomotopy

Nonlinearity, proper actions and equivariant stable cohomotopy

Stable Diffeomorphism Groups of~4-Manifolds

On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifolds

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