论文标题
拓扑复杂性，非球面和连接的总和
Topological complexity, asphericity and connected sums
论文作者
论文摘要
我们表明，如果一个封闭式$ n $ -manifold $ m $具有均匀度$ k $的非平凡的共同体学类别，它们的所有回调都可以撤回$ s^1 \ times n $消失的产品，那么拓扑复杂性$ \ mathrm {tc}（tc}（m）$至少是$ 6 $，如果$ n $是$ 7 $ 7 $ 7 $ 7 $ 7 $ 7 $ 7 $ 7 $ 7 $ 7。这些界限扩展并改善了Mescher的结果，例如将其申请到负弯曲的流形，并与至少一个这样的汇总进行连接的总和。实际上，由于Gromov Norm的不断变化，获得了更好的界限。结果，在第四维中，我们完全确定了这些连接的总和的拓扑复杂性，即表明它等于其最大值九。此外，我们讨论了Tori对两个同源性类别的实现，并展示了如何从可实现的类中构建不可实现的类别。本文的例子通常是非球面的歧管，其基本群体具有微不足道的中心和连接的总和。因此，我们讨论了最大拓扑复杂性$ 2N+1 $与非球门$ n $ manifolds中心及其连接的总和之间的微不足道之间的可能关系。
We show that if a closed oriented $n$-manifold $M$ has a non-trivial cohomology class of even degree $k$, whose all pullbacks to products of type $S^1\times N$ vanish, then the topological complexity $\mathrm{TC}(M)$ is at least $6$, if $n$ is odd, and at least $7$ or $9$, if $n$ is even. These bounds extend and improve a result of Mescher and apply for instance to negatively curved manifolds and to connected sums with at least one such summand. In fact, better bounds are obtained due to the non-vanishing of the Gromov norm. As a consequence, in dimension four, we completely determine the topological complexity of these connected sums, namely we show that it is equal to its maximum value nine. Furthermore, we discuss realisation of degree two homology classes by tori, and show how to construct non-realisable classes out of realisable classes in connected sums. The examples of this paper will quite often be aspherical manifolds whose fundamental groups have trivial center and connected sums. We thus discuss the possible relation between the maximum topological complexity $2n+1$ and the triviality of the center for aspherical $n$-manifolds and their connected sums.
