Seiberg-Witten Floer homotopy contact invariant

Abstract

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozváth-Szabó. Moreover, we prove a gluing formula relating our invariant with the first author's Bauer-Furuta type invariant, which refines Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary. As applications, we give two constraints for a certain class of symplectic fillings using equivariant K and KO-cohomology. We also treat the extension property of positive scalar curvature metrics on 4-manifolds with boundary.NOBUO IIDA AND MASAKI TANIGUCHI 5.2. Equivariant K theory 65 5.3. Extension problem of positive scalar curvature 70 References 72

Positive scalar curvature and homology cobordism invariants

“…However, the proof of (20) is the same as the proof of ( 21) in [15,Proposition 3.5]. So, we omit this.…”

“…In this subsection we prove the global slice theorem in our situation. We follow the method given in [15]. In [15], for 4-manifolds with conical end, a global slice theorem is given and the essentially same method can be applied to our situation.…”

“…The proof of the second statement is only difference between our construction and the usual Bauer-Furuta invariant. Here, we again follow the method given in [15]. To obtain the convergence of {x n }, we will use the following inequality obtained by the Fredholm property of L W [−∞,0] : there exists a constant C > 0 such that, for any x ∈ U k,δ ,…”

“…Here W is the 4-manifold defined by cutting X open along Y , and the 'left side' end is equipped with a periodic PSC metric and a neighborhood of the 'right side' boundary is equipped with a product metric of the form [0, 1] × Y . Technically, the relative Bauer-Furuta invariant over such a non-compact 4-manifold is defined using the similar method given in [15] which defines the relative Bauer-Furuta invariant for a certain class of 4-manifolds with conical end.…”

An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology

“…Although, we imposed the conditions b + 2 (X i ) ≡ 3 mod 4 in Theorem 1.23, if Y satisfies some nice conditions, we can also treat symplectic 4-manifolds with b + 2 (X i ) ≡ 1 mod 4. To this end we use the gluing result proved in [31, Theorem 1.2] relating Iida's invariant (1), the relative Bauer-Furuta invariant [32,46], and Seiberg-Witten Floer homotopy contact invariant [31]. We now review the relative Bauer-Furuta invariant.…”