Lower-bound conjecture for the moduli group of positive scalar curvature metrics

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Let MM be a closed spin manifold with fundamental group π1(M)=Γ\pi_1(M)=\Gamma and dimension dim⁡M=2k+1≥5\dim M=2k+1\geq 5, and suppose that MM carries a positive scalar curvature metric. Let P~(M)\widetilde P(M) be the coinvariant moduli group of concordance classes of positive scalar curvature metrics. Define Nfin(Γ)N_{\mathrm{fin}}(\Gamma) to be the cardinality of the set of positive integers that occur as orders of nontrivial elements of Γ\Gamma:

Nfin(Γ)=∣{d∈N+∣∃γ∈Γ with order⁡(γ)=d and γ≠e}∣.N_{\mathrm{fin}}(\Gamma)=\left|\left\{d\in\mathbb N_+\mid \exists\gamma\in\Gamma\text{ with }\operatorname{order}(\gamma)=d\text{ and }\gamma\neq e\right\}\right|.

Moduli-group rank conjecture. The rank of P~(M)\widetilde P(M) is at least Nfin(Γ)N_{\mathrm{fin}}(\Gamma).

This would give a lower bound for the size of the moduli space of positive scalar curvature metrics after quotienting by diffeomorphisms. The supplied source does not state whether the conjecture has been resolved.

References

Primary source

Zhizhang Xie and Guoliang Yu, “Higher rho invariants and the moduli space of positive scalar curvature metrics”, arXiv:1310.1136 (2014).

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