Nilpotency of the Bauer–Furuta stable homotopy Seiberg–Witten invariants

Furuta, Mikio; Kametani, Yukio; Minami, Norihiko

doi:10.2140/gtm.2007.10.147

Cited by 5 publications

(8 citation statements)

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“…We remark that, for any closed 4-manifold X with b + (X) ≥ 1 and b 1 (X) = 0, there is some large integer N such that, for any n ≥ N , the n-fold connected sum of X with itself, has a trivial stable cohomotopy Seiberg-Witten invariant. See [19] for more detail.…”

Section: Discussionmentioning

confidence: 99%

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number II: Applications

Ishida

Sasahira

2017

Communications in Analysis and Geometry

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We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant [4,5] of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a partial, but strong affirmative answer to a conjecture concerning non-vanishing of the invariant. Various new applications of the non-vanishing theorem are also given. For example, we shall introduce variants λ k of Perelman's λ invariants for real numbers k and compute the values for a large class of 4-manifolds including connected sums of certain Kähler surfaces. The non-vanishing theorem is also used to construct the first examples of 4-manifolds with non-zero simplicial volume and satisfying the strict Gromov-Hitchin-Thorpe inequality, but admitting infinitely many distinct smooth structures for which no compatible Einstein metric exists. Moreover, we are able to prove a new result on the existence of exotic smooth structures.where [X] is the fundamental class of X i and ·, · is the pairing between cohomology and homology.Notice that the image of c 1 (L Γ X ) under the natural map from H 2 (X, Z) to H 2 (X, Z 2 ) is equal to w 2 (X). This implies c 1 (L Γ X ) ∪ e i ∪ e j , [X] ≡ e i ∪ e j ∪ e i ∪ e j , [X] ≡ 0 mod 2.Hence the S ij (Γ X ) are integers.The first main theorem of this article can be stated as follows:

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Section: Discussionmentioning

confidence: 99%

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number II: Applications

Ishida

Sasahira

2017

Communications in Analysis and Geometry

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“…Then consider the following key commutative diagram, whose horizontal arrows are induced by the (N − 1)-fold iterated join map: Now, because of the Bauer-Furuta Seiberg-Witten invariants of a K3-surface with Spin-structure, we see {S(H 1 ), S( R 3 )} Pin 2 = ∅ Thus we apply the above key commutative diagram with k = 1, l = 3, N 0. Then, because of the Nilpotency Theorem [10], the bottom horizontal arrow is the trivial map for sufficiently large N (actually, straight-forward computations show this is trivial for N ≥ 5). Therefore, the right vertical map hits the constant map.…”

Section: Nilpotency Rules!mentioning

confidence: 99%

“…In this section we review the concept of level for free Z/2-spaces, and recall the results of Stolz [19], Furuta [5], Furuta-Kametani [7] and others in this terminology. Definition 2.1 (see Dai-Lam [3] and Dai-Lam-Peng [4]) For a free Z/2-space X, define the level of X, which we denote by level(X), as follows:…”

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confidence: 99%

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Homotopy theoretical considerations of the Bauer–Furuta stable homotopy Seiberg–Witten invariants

Furuta¹,

Kametani²,

Matsue³

et al. 2007

Geometry and Topology Monographs

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“…It turns out, however, that this is practically useless, except for manifolds Y with b + (Y ) = 0, such as Y = CP 2 , since the T-equivariant Bauer-Furuta invariants are known to be nilpotent otherwise. See [FKM07], for example. And clearly, if f is nilpotent, then the localisation with respect to f leads to trivial groups.…”

Section: Localisationmentioning

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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Nilpotency of the Bauer–Furuta stable homotopy Seiberg–Witten invariants

Abstract: We prove a nilpotency theorem for the Bauer-Furuta stable homotopy Seiberg-Witten invariants for smooth closed 4-manifolds with trivial first Betti number.

Cited by 5 publications

References 9 publications

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number II: Applications

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number II: Applications

Homotopy theoretical considerations of the Bauer–Furuta stable homotopy Seiberg–Witten invariants

Stable Diffeomorphism Groups of~4-Manifolds

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