关于$[3]*K$的可嵌入性
On the embeddability of $[3]*K$
摘要：
我们将单纯复形$[3]*K$嵌入到$\mathbb{R}^{n+2}$中的可嵌入性与$K$嵌入到$\mathbb{R}^n$中的可嵌入性联系起来。简而言之，在亚稳范围$2n\geq 3(d+1)$中，$K$到$\mathbb{R}^n$中的嵌入性等同于$[3]*K$到$\mathbb{R}^{n+2}$中的嵌入性。我们还证明了$K$的van Kampen阻塞消失当且仅当$[3]*K$的van Kampen阻塞消失。因此，对于$d=2$，$[3]*K$的嵌入性等同于$K$的van Kampen阻塞消失，但不等同于$K$的嵌入性。
英文摘要：
We relate the embeddability of the simplicial complex $[3]*K$ into $\mathbb{R}^{n+2}$ to that of $K$ into $\mathbb{R}^n$. In brief, the embeddability of $K$ into $\mathbb{R}^n$, in the metastable range $2n\geq 3(d+1)$, is equivalent to the embeddability of $[3]*K$ into $\mathbb{R}^{n+2}$. We show moreover than the van Kampen obstruction of $K$ vanishes if and only if the van Kampen obstruction of $[3]*K$ vanishes. It follows that for $d=2$, embeddabilty of $[3]*K$ is equivalent with the vanishing of the van Kampen obstruction for $K$, but not with the embeddability of $K$.
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