Curvature operators and rational cobordism (Q6639727)

In the article under review, for any given Riemannian manifold \((M, g)\) the authors define two real-valued functions \(C_1(R)\) and \(C_p(R)\) on \(M\), where \(R:\Lambda^2(TM)\to\Lambda^2(TM)\) is the curvature operator of \((M, g)\) and \(p\geq 2\) is an integer, in terms of highly nontrivial linear combinations of eigenvalues of \(R\) (and also of the largest eigenvalue of the Ricci curvature in the case of \(C_1(R)\)). The main results of the article under review are, under the assumption that \(M\) is spin, the curvature conditions \(C_1(R)>0\) and \(C_p(R)>0\) are preserved under surgeries of high codimensions and imply the vanishing of certain rational cobordism invariants, such as the twisted \(\widehat{A}\)-genera, Witten genus, elliptic genus, signature and all Pontryagin numbers. (Note that if \(C_p(R)>0\) for \(p\geq 2\), then the scalar curvature is positive.)\N\NIn the following we list two main results of the article under review. Let \((M, g)\) be a closed Riemannian spin \(n\)-manifold, where \(n=4k\) with \(k\geq 2\). The first main result states that for any parallel subbundle \(E\to M\) of the \(p\)-th tensor bundle \(TM^{\otimes p}\to M\), if \(C_p(R)>0\), then the twisted \(\widehat{A}\)-genera \(\widehat{A}(M, E\otimes\mathbb{C})\), defined by \(\widehat{A}(M, E\otimes\mathbb{C}):=\langle\widehat{A}(TM)\cup\textrm{ch}(E\otimes\mathbb{C}), [M]\rangle\), is zero. Moreover, for specific subbundles of \(TM^{\otimes p}\to M\), say \(E=\Lambda^p(TM)\) and \(E=S^p(TM)\), the authors give curvature conditions weaker than \(C_p(R)>0\) that still imply \(\widehat{A}(M, E\otimes\mathbb{C})=0\).\N\NAs a consequence of the first main result, suppose \(n\geq 24\) and \(n\neq 28\). If \(C_p(R)>0\) for \(p\geq 2\) and the first Pontryagin class of \(M\) is zero, then the Witten genus of \(M\) is zero.\N\NThe second main result states that the curvature conditions \(C_1(R)>0\) and \(C_p(R)>0\) are preserved under surgeries of certain codimensions. For example, the curvature condition \(C_1(R)>0\) is preserved under surgeries of codimension at least 10. Moreover, every oriented cobordism class \([M]\) with \(\dim(M)\geq 10\) that is not a nontrivial torsion class is represented by a Riemannian manifold with \(C_1(R)>0\), and every spin cobordism class \([M]\) with \(\dim(M)\geq 10\) and \(\widehat{A}(M, TM\otimes\mathbb{C})=0\) has a multiple which is represented by a Riemannian spin manifold with \(C_1(R)>0\).