Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number, I: Non-vanishing theorem

Ishida, Masashi; Sasahira, Hirofumi

doi:10.1142/s0129167x15410049

Cited by 5 publications

(7 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, if n = 4, then b + 2 (X) ≡ 4 (mod 8). Furthermore, Ishida and Sasahira [19] extended the sufficient condition (2) to the case b 1 = 0. For interesting examples and applications of these results, the readers can consult, for example, [18], [3], [7], and [20].…”

Section: Proofmentioning

confidence: 99%

Geometrically simply connected 4–manifolds and stable cohomotopy Seiberg–Witten invariants

Yasui

2019

Geom. Topol.

View full text Add to dashboard Cite

We show that every positive definite closed 4-manifold with b + 2 > 1 and without 1-handles has a vanishing stable cohomotopy Seiberg-Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented 4-manifold with b + 2 ≡ 1 and b − 2 ≡ 1 (mod 4) and without 1-handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the 1-handle condition, we prove these results under more general conditions which are much easier to verify.

show abstract

Section: Proofmentioning

confidence: 99%

Geometrically simply connected 4–manifolds and stable cohomotopy Seiberg–Witten invariants

Yasui

2019

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…We thus have a spin c structure s n on X n satisfying c 1 (s n ) = K n , SW Xn (s n ) ≡ 1 (mod 2), and d Xn (s n ) = 0. Hence, we see that (X n , s n ) is BF admissible in the sense of Ishida and Sasahira [11,Definition 2], due to the assumption on (X, s). One can also check that (K3, t) is BF admissible, where (K3, t) denotes the K3 surface equipped with a spin c structure t with c 1 (t) = 0.…”

Section: Proofsmentioning

confidence: 90%

“…A similar idea was used by the third author [29] to impose constraints on geometrically simply connected 4-manifolds and, more generally, on 4-manifolds admitting a non-torsion second homology class represented by a 2-handle neighborhood. We remark that the b 1 = 0 condition of [29,Theorem 2.4] can be relaxed to conditions similar to Theorem 1.2 (and hence Corollaries 1.3 and 1.4) of this paper without changing the proof, except that the connected sum formula of [11] is used instead of the formula of [4].…”

Section: Proofsmentioning

confidence: 97%

“…This theorem is implicit in the proof of [29,Theorem 2.4], which states that any 4-manifold with b 1 = 0 satisfying the assumption of Theorem 2.7 admits no symplectic structure for at least one orientation. The proof for the b 1 = 0 case of Theorem 2.8 is identical with the proof of [29,Theorem 2.4], and the proof for the general case also is identical, except that the connected sum formula of [11] is used instead of the formula of [4].…”

Section: Theorem 28 ([29]mentioning

confidence: 98%

“…One can also check that (K3, t) is BF admissible, where (K3, t) denotes the K3 surface equipped with a spin c structure t with c 1 (t) = 0. By a result of Ishida and Sasahira [11, Theorem A] (see also [11,Theorem 23 and Proposition 14]) on the Bauer-Furuta invariant [3], we see that K n is a Bauer-Furuta basic class of Z := X n #K3, and thus a monopole class of Z due to [10, Proposition 6]. Now let α be a second homology class of the K3 surface represented by a smoothly embedded, closed, oriented surface of genus g > 1 satisfying α•α = 2g −2.…”

Section: Proofsmentioning

confidence: 99%

See 2 more Smart Citations

The simple type conjecture for mod 2 Seiberg-Witten invariants

Kato¹,

Nakamura²,

Yasui³

2020

Preprint

View full text Add to dashboard Cite

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg-Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg-Witten invariant for at least one orientation.

show abstract

Yamabe invariants and the $\mathrm{Pin}^-(2)$-monopole equations

Ishida

Matsuo

Nakamura

2016

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

Abstract. We compute the Yamabe invariants for a new infinite class of closed 4-dimensional manifolds by using a "twisted" version of the SeibergWitten equations, the Pin − (2)-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature. IntroductionThe Yamabe invariant is a diffeomorphism invariant of smooth manifolds, which arises from a variational problem for the total scalar curvature of Riemannian metrics. The Pin − (2)-monopole equations are a "twisted" version of the Seiberg-Witten equations. In this paper we will compute the Yamabe invariants for a new infinite class of closed 4-dimensional manifolds by using the Pin − (2)-monopole equations. We begin by recalling the Yamabe invariant. Let X be a closed, oriented, connected manifold of dim X = m ≥ 3, and M(X) the space of all smooth Riemannian metrics on X. For each metric g ∈ M(X), we denote by s g the scalar curvature and by dµ g the volume form. Then the normalised Einstein-Hilbert functional E X : M(X) → R is defined byThe classical Yamabe problem is to find a metricǧ in a given conformal class C such that the normalised Einstein-Hilbert functional attains its minimum on C: E X (ǧ) = inf g∈C E X (g). This minimising metricǧ is called a Yamabe metric, and a conformal invariant Y(X, C) := E X (ǧ) the Yamabe constant. We define a diffeomorphism invariant Y(X) by the supremum of Y(X, C) of all the conformal classes C on X:We call it the Yamabe invariant of X; it is also referred to as the σ-constant. See [16] and [28]. It is a natural problem to compute the Yamabe invariant. In dimension 4, Seiberg-Witten theory and LeBrun's curvature estimates have played a prominent role in this problem. LeBrun used the ordinary Seiberg-Witten equations to compute the Yamabe invariants of most algebraic surfaces [19,20]

show abstract

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number, I: Non-vanishing theorem

Cited by 5 publications

References 11 publications

Geometrically simply connected 4–manifolds and stable cohomotopy Seiberg–Witten invariants

Geometrically simply connected 4–manifolds and stable cohomotopy Seiberg–Witten invariants

The simple type conjecture for mod 2 Seiberg-Witten invariants

Yamabe invariants and the $\mathrm{Pin}^-(2)$-monopole equations

Contact Info

Product

Resources

About