Hspaces, loop spaces and the space of positive scalar curvature metrics on the sphere
Citation
Walsh, Mark. 2014. Hspaces, loop spaces and the space of positive scalar curvature metrics on the sphere. Geometry & Topology, vol. 18:no. 4:pp 2189–2243
Abstract
For dimensions n >= 3, we show that the space Riem(+)(Sn) of metrics of positive scalar curvature on the sphere Sn is homotopy equivalent to a subspace of itself which takes the form of an Hspace with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action on this subspace of the operad obtained by applying the bar construction to the little ndisks operad. Using results of Boardman, Vogt and May we show that this implies, when n >= 3, that the path component of Riem(+)(Sn) containing the round metric is weakly homotopy equivalent to an nfold loop space. Furthermore, we show that when n = 3 or n >= 5, the space Riem(+)(Sn) is weakly homotopy equivalent to an nfold loop space provided a conjecture of Botvinnik concerning positive scalar curvature concordance is resolved in the affirmative.
Description
Click on the DOI link to access the article for free at the publisher's website.
Collections
Related items
Showing items related by title, author, creator and subject.

Prioritizing scientific exploration: A comparison of the ethical justifications for space development and for space science
Schwartz, James S. J. (Elsevier Ltd., 201411)I argue that the moral justification for space science is more compelling than the moral justification for space development. Thus, we ought to reemphasize the status of science as a major stakeholder in space, especially ... 
Kdistributed vector random fields in space and time
Ma, Chunsheng (Elsevier, 201304)This paper introduces two types of secondorder vector random fields or stochastic processes whose marginals are Kdistributed, through certain mixture procedures. The first type is formulated as an independent product of ... 
Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics
Walsh, Mark (American Mathematical Society, 201307)Let X and Y be a pair of smooth manifolds, each obtainable from the other by surgery in codimension at least three. We show that the corresponding spaces Riem(+)(X) and Riem(+)(Y), respectively consisting of Riemannian ...