关于完全非自旋流形的Gromov猜想
On Gromov's conjecture for totally non-spin manifolds
摘要：
Gromov猜想指出，对于一个具有正数量曲率的闭$n$-流形$M$，其泛覆盖$\tilde M$的宏观维数满足不等式$\dim_{mc}\tilde M\le n-2$ \cite{G2}。对于基本群是具有$vcd\ne n$的虚对偶群的完全非自旋$n$-流形，我们证明了这个不等式。在虚拟交换群的情况下，我们减少格罗莫夫的猜想完全非自旋流形的消失问题是否$H_n(T^n)^+= 0$的$n$-环面$T^n$其中$H_n(T^n)^+\subset H_n(T^n)$是子群的同源类，可以实现的流形具有正的标量曲率。
英文摘要：
Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.
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