具有相对2-骨架的正规4-伪流形
Normal $4$-pseudomanifolds with a relative 2-skeleton
AI总结：
本文研究正规4-伪流形在g₂和g₃最优条件下的结构，证明其可由单形边界复形经顶点折叠、边折叠、连通和及单顶点悬浮等操作序列生成。
摘要：
单纯复形中与面数相关的不变量的研究是组合拓扑学的核心课题。其中，不变量$g_2$扮演着重要角色。对于正规$d$-伪流形$K$（$d \geq 3$），已知对每个顶点$v$都有$g_2(K) \geq g_2(lk(v, K))$。若$K$至多有两个奇点，且对某个奇点$t$满足$g_2(K) = g_2(lk(t, K))$，则$g_3(K) \geq g_3(lk(t,K))$成立。若正规$d$-伪流形$K$对某个奇点$t$满足$g_2(K) = g_2(lk(t,K))$且$g_3(K) = g_3(lk(t,K))$，则称$K$为$g_2$-最优和$g_3$-最优的。本文在$g_2$-最优和$g_3$-最优条件下建立了正规$4$-伪流形的结构性结果。我们证明：若$K$是恰好有一个奇点$t$的正规$4$-伪流形，且在$t$处是$g_2$-最优和$g_3$-最优的，则$K$可以通过一系列顶点折叠型和连通和型操作从$5$-单形的边界复形得到。当$K$恰好有两个奇点，且在其中一个奇点处是$g_2$-最优和$g_3$-最优时，它可以通过一系列单顶点悬浮型、顶点折叠型和连通和型操作从$4$-单形的边界复形得到。另一方面，我们证明：若$K$有两个奇点，且在其中一个奇点处是$g_2$-最优和$g_3$-最优的，则它可以通过一系列顶点折叠型、边折叠型和连通和型操作从$5$-单形的边界复形得到。
英文摘要：
The study of face-number-related invariants in simplicial complexes is a central topic in combinatorial topology. Among these, the invariant $g_2$ plays a significant role. For a normal $d$-pseudomanifold $K$ ($d \geq 3$), it is known that $g_2(K) \geq g_2(lk(v, K))$ for every vertex $v$. If $K$ has at most two singularities and satisfies $g_2(K) = g_2(lk(t, K))$ for a singular vertex $t$, then $g_3(K) \geq g_3(lk(t,K))$ holds. A normal $d$-pseudomanifold $K$ is called $g_2$- and $g_3$-optimal if $g_2(K) = g_2(lk (t,K))$ and $g_3(K) = g_3(lk (t,K))$ for a singular vertex $t$. In this article, we establish structural results for normal $4$-pseudomanifolds under $g_2$- and $g_3$-optimality conditions. We show that if $K$ is a normal $4$-pseudomanifold with exactly one singular vertex $t$ and is $g_2$- and $g_3$-optimal at $t$, then $K$ can be obtained from boundary complexes of $5$-simplices through a sequence of operations of types vertex foldings and connected sums. When $K$ has exactly two singularities and is $g_2$- and $g_3$-optimal at one singular vertex, it is derived from the boundary complexes of $4$-simplices through a sequence of operations of types one-vertex suspensions, vertex foldings, and connected sums. Alternatively, we prove that if $K$ has two singular vertices and is $g_2$- and $g_3$-optimal at one of them, then it arises from boundary complexes of $5$-simplices through a sequence of operations of types vertex foldings, edge foldings, and connected sums.
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