Moduli space of nonnegatively curved metrics on manifolds of dimension 4k + 1

Spaces of positive scalar curvature metrics on totally nonspin manifolds with spin boundary

“…The manifold β however is not oriented nullbordant, since the α-invariant of a spin manifold in these dimensions is equal to a multiple of its Â-genus, which is a Pontryagin number, hence it is not possible to use these rigidity results. 7 However, when working with concordance classes instead, we have the following result pointing towards an affirmative answer of Question 4.1. Theorem 4.2 Let M be a d-dimensional, closed, oriented, totally nonspin manifold with D ⊂ M an embedded disk and let g ∈ R + (M \ int(D)) h • .…”

“…In the case where M is nonspin, but its universal cover is Spin, there are cases where R + (M) is not connected or has even infinitely many path components, see [2,7,8,15,16,23,25]. For totally nonspin manifolds, Kastenholz-Reinhold give an example of a closed, totally nonspin manifold of dimension 6 whose space of psc-metrics has infinitely many components in [21].…”

Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics

“…The codimension 3 case requires a bit more work. Recall that if M is the total space of a linear S 3 -bundle over S 4 , then M admits a metric of K 0 [23], and moreover, if the bundle has nonzero Euler number, then M K 0 (M ) has infinitely many connected components [12,21]. We prove: Theorem 1.4.…”

“…Examples of closed manifolds S for which M K 0 (S) has infinitely many connected components can be found in [12,13,14,15,21,25]. These metrics on S have K 0 and scal > 0, and the connected components are distinguished by index-theoretic invariants that are constant on paths of scal > 0.…”

“…To prove Theorem 1.4 we use results of Grove-Ziller [23] and some topological considerations to find an indecomposable V with a codimension three soul, and then we observe that the metric on the soul can be moved by Cheeger deformation to metrics in [12,21] that represent infinitely many connected components.…”