\(T\)-structure and the Yamabe invariant (Q2880484)

The Yamabe invariant, introduced by the author [Differ. Geom. Appl. 24, No. 3, 271--287 (2006; Zbl 1099.53031); J. Geom. Phys. 59, No. 2, 246--255 (2009; Zbl 1163.53023)], is an invariant of a smooth closed manifold depending on its smooth topology. Let \(M\) be a smooth closed manifold of dimension \(n\). Given a smooth Riemannian metric \(g\) on it, the conformal class \([g]\) is defined as NEWLINE\([g] = \{\varphi g \mid \varphi:M \rightarrow \mathbb{R}^{+}text{ is smooth}\}\). The famous Yamabe problem [\textit{J. Lee} and \textit{T. Parker}, ``The Yamabe problem'', Bull. Am. Math. Soc., New Ser. 17, 37--91 (1987; Zbl 0633.53062)] states that there exists a metric \(\widetilde{g}\) in \([g]\) which attains the minimum NEWLINE\[NEWLINE \inf _{\widetilde{g}\in [g]}\frac{\int_M s_{\widetilde{g}}dV_{\widetilde{g}}}{(\int_M dV_{\widetilde{g}})^ {\frac{n-2}{n}}},NEWLINE\]NEWLINE where \(s_{\widetilde{g}}\) and \(dV_{\widetilde{g}}\) denote the scalar curvature and the volume element of \(\widetilde{g}\). NEWLINENEWLINENEWLINE NEWLINEIn turns out that when \(n\geq 3\), a unit-volume minimizer \(\widetilde{g}\) in \([g]\) has constant scalar curvature, which is equal to the above minimum value called the Yamabe constant of \([g]\) and denoted by \(Y(M,[g])\). It is known that the Yamabe constant of any \(n\)-manifold is bounded above by \(Y(S^n,[g_0])\) where \([g_0]\) denotes a standard round metric. Thus following a min-max procedure the author defines the Yamabe invariant NEWLINE\[NEWLINEY(M):=\sup_{[g]}Y(M,[g])NEWLINE\]NEWLINE on \(M\). The Yamabe invariant contains information about possible scalar curvature on it. For example:NEWLINENEWLINE\(\bullet\) \(Y(M)>0\) if and only if \(M\) admits a metric of positive scalar curvature.NEWLINENEWLINE\(\bullet\) If \(M\) is simply-connected and dim \(M\geq 5\), then \(Y(M)\geq 0\). With the further assumption that \(M\) is spin, \(Y>0\) if and only if the \(\alpha\)-genus of \(M\) is 0, etc. In particular, it is well-known that a product manifold \(T^m\times B\) where \(T^m\) is the m-dimensional torus, and \(B\) is a closed spin manifold with nonzero \(\widehat{A}\)-genus has zero Yamabe invariant.NEWLINENEWLINEIn this paper, the author generalizes this to various \(T\)-structured manifolds (in the sense of \textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 23, 309--346 (1986; Zbl 0606.53028); J. Differ. Geom. 32, No. 1, 269--298 (1990; Zbl 0727.53043)], for example \(T^m\)-bundles over such \(B\) whose transition functions take values in \(Sp(m,\mathbb{Z})\) (or \(Sp(m-1,\mathbb{Z})\oplus\{\pm 1\}\) for odd \(m\)).NEWLINENEWLINEThe main results are the following:NEWLINENEWLINETheorem 4.1. Let \(B\) be a closed spin manifold of dimension 4d with nonzero \(\widehat{A}\)-genus, and \(M\) be a \(T^m\)-bundle over \(B\) whose transition functions take values in \(Sp(m,\mathbb{Z})\) (or \(Sp(m-1),Z)\oplus\{\pm 1\}\) for odd \(m\)). Then \(Y(m)=0\).NEWLINENEWLINETheorem 4.5. Let \(B\) be a closed spin manifold of dimension \(4d\) with nonzero \(\widehat{A}\)-genus, and \(M\) be an \(S^1\) or \(T^2\)-bundle over \(B\) whose transition functions take values in \(GL(1,\mathbb{Z})\) or \(GL(2,\mathbb{Z})\) respectively. Then \(Y(M)=0\).NEWLINENEWLINECorollary 4.8. Let \(M\) be a \(T^m\)-bundle in all the above so that \(Y(M)=0\). If \(\dim M=4n\), then NEWLINE\[NEWLINEY(M \sharp k \mathbb{H}P^n\sharp l \overline{\mathbb{H}P^n})=0,NEWLINE\]NEWLINE and if \(\dim M = 16\), then NEWLINE\[NEWLINEY(M\sharp k \mathbb{H}P^4 \sharp l \overline{\mathbb{H}P^4} \sharp k' Ca P^2 \sharp l' \overline{CaP^2})=0,NEWLINE\]NEWLINE where \(k,l,k'\) and \(l'\) are nonnegative integers, and the overline denotes the reversed orientation.NEWLINENEWLINETheorem 4.9. Let \(B\) be a closed oriented manifold of dimension \(\leq 3\) , and \(X\) be an \(S^1\) or \(T^2\)-bundle over \(B\). Suppose that \(X\times T^m\) for \(m=4-\dim X\) has a finite cover \(M\) with \(b^+_2(M)>1\) which is a \(T^2\)-bundle over an oriented surface whose transition functions take values in a discrete subset of \(T^2\rtimes SL(2,\mathbb{Z})\). Then \(Y(X)=0\)