It is proved that the suspension of a closed n-dimensional manifold M, $n\ge1$, does not embed in a product of n+1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a far-reaching generalization the Borsuk theorem on non-embeddability of the (n+1)-dimensional sphere in a product of n+1 curves.
The telescope approach to embeddability of compacta
We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - Δadmits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the factor-exchanging involution on X x X - Δis trivial. Assuming that X quasi-embeds in R^{2n} (i.e. is an inverse limit of n-polyhedra, embeddable in R^{2n}), this is equivalent to the vanishing of an obstruction in lim^1 H^{2n-1}(K_i) over compact subsets K_i \subset X x X - Δ. One application is that an n-dimensional ANR embeds in R^{2n} if it quasi-embeds in R^{2n-1…
Embeddability of arrangements of pseudocircles and graphs on surfaces
伪圆是曲面上的一条简单闭曲线;伪圆的排列是一组伪圆的集合，这些伪圆两两相交于它们相交的两点。Ortner证明，安排pseudocircles是嵌入到球当且仅当其所有的子安排的大小最多4是嵌入到球，并要求如果一个类似的结果成立嵌入到可定向曲面的高等属。我们肯定地回答了这个问题：伪圆的排列是可嵌入到亏格~ $g$的可定向曲面中的当且仅当它的所有大小至多为$4g+4$的子排列都是可嵌入的。而且，这个界限很紧。实际上，我们对一个更一般的排列概念也有类似的结果，我们称之为\emph{arrangement of graphs}。
Free cyclic actions on surfaces and the Borsuk-Ulam theorem
设$M$和$N$是拓扑空间，$G$是群，$τ\colon\thinspace G \times M \to M$是$G$的真自由作用.本文定义了从$M$到$N$关于$(G,τ)$的同伦类的Borsuk-Ulam型性质，推广了从$n$ -球面$\mathbb{S}^n$到$\mathbb{R}^n$的经典对映Borsuk-Ulam定理.在$M$是有限路径连通CW-复形，$G$是有限非平凡阿贝尔群，$τ$是真自由胞腔作用，$N$是$\mathbb{R}^2$或$\mathbb{S}^2$与$\mathbb{RP}^2$无边界差的紧致曲面的情形下，给出了一个包含辫群的代数判据，用以判定自由同伦类$β\in [M,N]$是否具有Borsuk-Ulam性质.作为这个准则的应用，我们考虑了$M$是一个无边界的紧致曲面，它具有有限循环群$\mathbb{Z}_n$的自由作用$τ$的情形.利用$M$的轨道空间$M_τ$通过作用$τ$的可定向性、$n$模$4$的值以及涉及$M_τ$的第一同调群的一定代数条件，我们能够确定从$M$到$\mathbb{R}^2$的单同伦类映射是否具有关于$(\mathbb{Z}_n,τ)$的Borsuk-Ulam性质.最后，我们给出了对称群作用于曲面的一些例子，并得到了以$\mathbb{R}^2$为目标的映射的Borsuk-Ulam性质的部分结果.
