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Let $\mathcal{R}_{+\mathrm{sc}}(S^n)$ denote the space of complete Riemannian metrics of positive scalar curvature on the sphere $S^n$. It's known that $\mathcal{R}_{+\mathrm{sc}}(S^2)$ is contractible (Rosenberg-Stolz), that $\mathcal{R}_{+\mathrm{sc}}(S^3)$ is connected (Codá Marques), and that $\mathcal{R}_{+\mathrm{sc}}(S^{4n-1})$, $n\geq2$, has infinitely many components (Carr). Hitchin has also shown that $\mathcal{R}_{+\mathrm{sc}}(S^{n})$ is not connected when $n\geq 9$ and $n\equiv 0,1\;(8)$ (if I recall the figures correctly...)

What is known about the space $\mathcal{R}_{+\mathrm{sc}}(S^4)$?

More than just path components, I'm interested in the higher homotopy (and other algebraic-topological features) of this space. In a fairly recent paper M. Walsh seems to suggest that he believes that the space is not path connected, and proves at least that $\pi_1\mathcal{R}_{+sc}(S^4)$ is abelian. I'm not sure if more is known.

There is also the moduli space $\mathcal{M}_{+sc}(S^n)=\mathcal{R}_{+\mathrm{sc}}(S^n)/\mathrm{Diff}_{e_1}(S^n)$ to consider. This space has been studied by several authors, but most of the results I have seen apply to large $n$, and not the $n=4$ case in which I am most interested.

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    $\begingroup$ It seems the questions you ask are open problems of current interest, and hence are not appropriate for this site. $\endgroup$ Nov 19, 2019 at 13:46

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