The positive scalar curvature cobordism category

TL;DR

The paper develops a robust framework tying spaces of positive scalar curvature metrics to infinite loop spaces via a PSC cobordism category and Segal's Gamma-space machinery. Central to the approach are fibre theorems for forgetful functors, PSC surgery arguments, and concordance categories that identify spaces of PSC metrics as loop- or infinite-loop spaces under suitable connectivity hypotheses (notably $d\ge 6$). It also analyzes the action of diffeomorphism groups on PSC metrics, proving that such actions factor through the Madsen–Tillmann spectrum and yield rigidity results when rational Pontrjagin classes vanish, together with delooped index-theoretic consequences for spin manifolds. An index-difference theory is delooped to KO-theory, enabling infinite-loop-map properties and new proofs of prior results, while the concordance-implies-isotopy conjecture is discussed for potential simplifications. In sum, the paper provides a cohesive, high-dimensional homotopy-theoretic account of PSC metrics, their moduli, and their index-theoretic and diffeomorphism-group symmetries, producing new infinite-loop-structure results and linking geometric analysis with stable homotopy theory.

Abstract

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fibre of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen's Theorem B and instances of the Gromov--Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second named author to the psc setting to pass between different connectivity conditions. Segal's theory of $Γ$-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen--Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontrjagin classes. A delooped version of the Atiyah--Singer index theorem proved by the first named author is used to moreover show that the secondary index invariant to real $K$-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

Figures (4)

• Figure 1: A point in $((F^{2,1}/M)/L)_{0,0}$, which is sent by $\zeta^L_{0,0}$ to $(W,s, h\vert_N) \in \mathrm{Ob}(F^{2,1}/M)$. A path in $\mathrm{Ob}(F^{2,1}/M)$ starting at this point which immediately deforms the middle of $W\vert_{[s-(b-t),s]}$ to be non-cylindrical cannot be lifted, as a lift would have to vary $t$ continuously and hence must remain cylindrical over a slightly smaller interval.
• Figure 2: The manifold $K$ with $d=2$ and $l=0$.
• Figure 3: The manifolds (a) $\tilde{K}$, and (b) $K\vert_{(-6,0)}$, with $d=2$ and $l=0$.
• Figure 4: The manifold $U$, with the $S^l$-direction not drawn.

Theorems & Definitions (197)

• Theorem A
• Theorem B
• Remark 1.1.1
• Theorem C
• Theorem D
• Theorem E
• Remark 1.3.1
• Theorem F
• Remark 1.3.2
• Theorem G