[Paper Review] H-Space structures on spaces of metrics of positive scalar curvature

Georg Frenck|arXiv (Cornell University)|Apr 2, 2020

Advanced Differential Geometry Research10 references4 citations

TL;DR

This paper constructs a geometric H-space multiplication on the space of positive scalar curvature metrics on spin manifolds that are nullcobordant in their own tangential 2-type. Using a graphical calculus based on relative Spin×Bπ-cobordism classes, it establishes that this multiplication is homotopy-associative and homotopy-commutative, and applies the structure to prove a rigidity criterion for the diffeomorphism group action on the space of psc-metrics, with a key result showing an H-space equivalence between the cylinder space and the double of a disk in high dimensions.

ABSTRACT

We construct and study an $H$-space multiplication on $\mathcal R^+(M)$ for manifolds $M$ which are nullcobordant in their own tangential $2$-type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on $\mathcal R^+(M)$ via pullback. We also compare this to other known multiplicative structures on $\mathcal R^+(M)$.

Motivation & Objective

• To construct a geometric H-space multiplication on the space of positive scalar curvature metrics on spin manifolds that are nullcobordant in their tangential 2-type.
• To establish a rigidity criterion for the action of the diffeomorphism group on the space of psc-metrics via this H-space structure.
• To compare the new H-space structure with existing multiplicative structures on the space of psc-metrics.
• To investigate the independence of the H-space structure on the choice of nullcobordism and its implications for homotopy equivalences between metric spaces.
• To explore the existence of stable metrics and H-space equivalences in the context of doubles of nullcobordisms, particularly in the case of spheres and disks.

Proposed method

• The construction uses a relative cobordism class in Ω^Spin,π_d(M₀ ∐ M₁, M) to define a map S(X_W): R⁺(M) × R⁺(M) → R⁺(M), which induces the H-space multiplication.
• Graphical calculus is developed to compute in the cobordism set, enabling explicit verification of homotopy associativity and commutativity.
• The H-space structure is defined via a surgery map S(W) associated to a nullcobordism W of a manifold M, with the multiplication given by gluing two copies of the opposite cobordism and one copy of W.
• The construction relies on the parametrized Gromov–Lawson–Schoen–Yau surgery theorem to ensure weak homotopy equivalences in metric space inclusions.
• The paper uses the existence of right-stable and left-stable metrics on nullcobordisms to construct homotopy equivalences between metric spaces of cylinders and doubles.
• The key technical tool is the map cl_{G_rst}, which glues in a stable metric on a nullcobordism to produce a homotopy equivalence from a cylinder space to a double space.

Experimental results

Research questions

• RQ1 Does the H-space structure on R⁺(M) constructed via a nullcobordism W depend on the choice of W, and if so, how are different choices related?
• RQ2 Can the action of the diffeomorphism group on R⁺(M) be shown to be trivial in the homotopy category using the H-space structure?
• RQ3 Is there a natural H-space equivalence between the space of metrics on a cylinder over M and the space of metrics on the double of a nullcobordism?
• RQ4 Under what conditions does the inclusion of metric spaces with boundary conditions induce a weak homotopy equivalence?
• RQ5 Can the H-space structure on R⁺(S^{d-1}) induced by the standard disk be shown to be equivalent to the H-space structure on the cylinder space R⁺(S^{d-2}×[0,1])?

Key findings

• The space R⁺(M) of positive scalar curvature metrics on a spin manifold M of dimension ≥6 that is Spin×Bπ₁(M)-nullcobordant admits a homotopy-associative, homotopy-commutative H-space structure.
• The H-space multiplication is geometrically defined via a surgery map associated to a nullcobordism W, and the resulting map μ_W: R⁺(M) × R⁺(M) → R⁺(M) is independent of the choice of W up to H-space equivalence.
• For simply connected spin manifolds M of dimension ≥6, the action of the oriented diffeomorphism group on R⁺(M) is trivial in the homotopy category if the manifold is nullcobordant in its tangential 2-type.
• The map cl_{g_tor}: (R⁺(S^{d-2}×[0,1])_{g₀,g₀}, μ_cyl) → (R⁺(S^{d-1}), μ_D) is an H-space equivalence when d ≥ 7.
• The H-space structure on R⁺(S^{d-1}) induced by the standard disk D^d is equivalent to the H-space structure on the cylinder space R⁺(S^{d-2}×[0,1])_{g₀,g₀} via the gluing map cl_{g_tor}.
• The H-space structure on R⁺(M) is preserved under homotopy equivalences induced by cobordisms with non-vanishing α-invariant, but such maps are not homotopic to the identity.