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Let $M^n$ be a compact simply connected spin manifold.

Gromov, Lawson, and Stolz proved that if $ngeq 5$, then $M$ admits a metric of positive scalar curvature iff $alpha(M)=0$.

Question: What happens in dimension 4?

1. Are there compact simply connected spin manifolds of dimension 4 which have a metric of positive scalar curvature and $alpha(M)neq 0$?


2. Are there compact simply connected spin manifolds of dimension 4 which have $alpha(M)=0$ but no metric of positive scalar curvature?


3. What are necessary and/or sufficient conditions for a compact simply connected spin 4-manifold to have a metric with positive scalar curvature?

1. Lichnerowicz proved that if $M$ is a closed spin manifold which admits a positive scalar curvature metric, then $hatA(M) = 0$. In dimensions $4k$, $alpha(M) = 2hatA(M)$, so it follows that $alpha(M) = 0$. In fact, Hitchin proved that $alpha(M) = 0$, regardless of dimension.



2. 
   

   In dimension four, $hatA(M) = -frac124p_1(M) = -frac18tau(M)$, so $alpha(M) = 0$ if and only if $tau(M) = 0$. If $M$ is a simply connected, closed, spin four-manifold with signature zero, then by Freedman's classification, there is an integer $k geq 0$ such that $M$ is homeomorphic to $k(S^2times S^2)$, the connected sum of $k$ copies of $S^2times S^2$; note, the connected sum of zero copies of $S^2times S^2$ is defined to be $S^4$. If $M$ is diffeomorphic to $k(S^2times S^2)$, then $M$ admits positive scalar curvature metrics. Said another way, the topological manifolds $k(S^2times S^2)$ all admit positive scalar curvature for their standard smooth structure.

   
   
   

   A closed four-manifold can potentially admit a countably infinite number of smooth structures, and the existence of a positive scalar curvature metric depends on which smooth structure you choose. So it is possible that for some choice of $k$ and some choice of smooth structure on $k(S^2times S^2)$, there does not admit a positive scalar curvature metric. Such examples exist. In Hirzebruch Surfaces: Degenerations,
   Related Braid Monodromy, Galois Covers, Teichner constructs examples of simply connected general-type complex surfaces which are spin and have signature zero; see Theorem $5.8$. These are homeomorphic to $k(S^2times S^2)$ for some $k$, but general type surfaces do not admit positive scalar curvature metrics.

   
   



3. In dimension four, unlike in other dimensions, there are gauge theoretic obstructions to the existence of positive scalar curvature metrics. In particular, on a closed smooth four-manifold with $b^+ geq 2$, the existence of a positive scalar curvature metric implies that all of its Seiberg-Witten invariants vanish. This is a necessary condition, but it is not sufficient. Note that these invariants are usually too complicated to calculate, except in ideal situations such as when the four-manifold in question is the underlying four-manifold of a Kähler surface.