On the topology of the space of Ricci-positive metrics

Abstract

We show that the space R pRc (W 2n g ) of metrics with positive Ricci curvature on the manifold W 2n g := ♯ g (S n × S n ) has nontrivial rational homology if n ≡ 3 (mod 4) and g are both sufficiently large. The same argument applies to R pRc (W 2n g ♯N ) provided that N is spin and W 2n g ♯N admits a Ricci positive metric.

Mapping class groups of highly connected $$(4k+2)$$-manifolds

“…) (so in particular to n 1,1/2 ) using the flip of the factors and a diffeomorphism of S n of degree −1. 5 In contrast to the case n ≥ 3, the resulting analogues of the morphisms (4.2) and (4.4) for n ≥ 4 even are isomorphisms for all g ≥ 1; this is Theorem G. 5 There are several mistakes in the literature related to the fact that G 1 = O 1,1 (Z) is isomorphic to (Z/2) 2 and not Z/4: in [35, p. 645], it should beπ 0 Diff(S 2 × S 2 ) ∼ = (Z/2) 2 , in [49,Thm 1] it should bẽ π 0 Diff(S p × S q )/π 0 SDiff(S p × S q ) ∼ = (Z/2) 2 for p = q even, and finally in the proof of [36,Thm 2.6] it should be Aut(H k (S k × S k )) ∼ = (Z/2) 2 for k even.…”

“…The description of n g up to these two extension problems has found a variety of applications [2,[5][6][7]18,23,29,33,38,39], especially in relation to the study of moduli spaces of manifolds [22]. The remaining extensions (1) and (2) have been studied more closely for particular values of g and n [15,19,21,36,37,48] but are generally not well-understood (see e.g.…”

AnHP2-bundle overS4with nontrivial Â-genus

“…The standard Riemannian metric g st on HP 2 has positive sectional curvature, so pulling back g st along orientation-preserving diffeomorphisms yields a map (−) * g st : Diff(HP 2 ) −→ R sec>0 (HP 2 ) ⊂ R Ric>0 (HP 2 ) ⊂ R scal>0 (HP 2 ) from the group of diffeomorphisms of HP 2 in the smooth topology to the spaces of Riemannian metrics on HP 2 having positive sectional, Ricci, or scalar curvature. By an argument of Hitchin [10] (see for instance [1, p. 3999] for an explanation of this), the theorem has the following corollary, which answers a question of Schick [6, p. 30] and provides an example as asked for in [1,Remark 2.2].…”

“…If one is willing to instead consider HP n -bundles over S 4m for n sufficiently large compared with m, then one may replace the appeal to the Lemma by the more classical [3,Corollary D], which implies that the map π 4m (hAut(HP n )/ Diff(HP n ); id) ⊗ Z 1 2 −→ π 4m (hAut(HP n )/ Diff(HP n ); id) ⊗ Z 1 2 is (split) surjective as long as 4m − 1 lies in the pseudoisotopy stable range for HP n (so 3m < n suffices, by [11]). See also [2, Theorem 1].…”

Moduli spaces of manifolds: a user's guide

“…[Kup16] to establish the finite generation of homotopy groups of Diff ∂ (D d ) for d = 4, 5, 7. (ii) These results have been used by Botvinnik, Ebert, and Randal-Williams[BERW17], Ebert and Randal-Williams[ERW17], and Botvinnik, Ebert, and Wraith[BEW18] to study the topology of spaces of Riemannian metrics of positive scalar, or Ricci, curvature. (iii) These results have been used by Krannich[Kra18] to show that if Σ is a homo-…”