Positive scalar curvature and higher‐dimensional families of Seiberg–Witten equations

Abstract

We introduce an invariant of tuples of commuting diffeomorphisms on a 4‐manifold using families of Seiberg–Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1‐parameter families of Seiberg–Witten equations. Our invariant yields an application to the homotopy groups of the space of positive scalar curvature metrics on a 4‐manifold. We also study the extension problem for families of 4‐manifolds using our invariant.

Families of diffeomorphisms and concordances detected by trivalent graphs

“…Namely, all classes given by surgery on trivalent graphs are in the image of the induced map 𝜄 * ∶ 𝜋 𝑞 ℳ 𝗉𝗌𝖼 𝜕 (𝑋) ℎ 0 → 𝜋 𝑞 𝐵Dif f 𝜕 (𝑋) for an arbitrary smooth manifold 𝑋 of dimension 𝑑 ⩾ 4 having a psc metric ℎ 0 . For 𝑑 = 4, Theorem 1.8 is in contrast to that the classes in 𝜋 𝑞 𝐵Dif f (𝑋) detected by Seiberg-Witten theory do not admit fiberwise psc-metrics [11,17].…”

“…Namely, all classes given by surgery on trivalent graphs are in the image of the induced map $$\iota _*:\pi _q\mathcal {M}_\partial ^\mathsf {psc}(X)_{h_0}\rightarrow \pi _qB\mathrm{Diff}_\partial (X)$$ for an arbitrary smooth manifold $$X$$ of dimension $$d\geqslant 4$$ having a psc metric $$h_0$$. For $$d=4$$, Theorem 1.8 is in contrast to that the classes in $$\pi _qB\mathrm{Diff}(X)$$ detected by Seiberg–Witten theory do not admit fiberwise psc‐metrics [11, 17].…”

Homological instability for moduli spaces of smooth 4-manifolds

“…This proves the claim (i). The remaining claims (ii), (iii) follow from the homotopy commutative diagram (24) with homotopy equivalences M x 0 pXq » BDiff x 0 pXq and M B p Xq » BDiff B p Xq.…”

“…Positive scalar curvature. The vanishing of the Seiberg-Witten invariant for a positive scalar curvature metric extends to invariants for families, and this generalization is useful to study the topology of spaces of positive scalar curvature metrics [50,24]. In this subsection, we formulate a corresponding vanishing theorem in our setup.…”