Small covers over a product of simplices

Abstract: Choi shows that there is a bijection between Davis–Januszkiewicz equivalence classes of small covers over an $n$-cube and the set of acyclic digraphs with $n$-labeled vertices. Using this, one can obtain a bijection between weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over an $n$-cube and the isomorphism classes of acyclic digraphs on labeled $n$ vertices up to local complementation and reordering vertices. To generalize these results to small covers over a product of simplices we introduce the notion of $\omega$-weighted digraphs for a given dimension function $\omega$. It turns out that there is a bijection between Davis–Januszkiewicz equivalence classes of small covers over a product of simplices and the set of acyclic $\omega$-weighted digraphs. After introducing the notion of an $\omega$-equivalence, we also show that there is a bijection between the weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over $\Delta^{n_1}\times\cdots \times \Delta^{n_k}$ and the set of $\omega$-equivalence classes of $\omega$-weighted digraphs with $k$-labeled vertices $\{v_1, \cdots, v_k\}$ where $\omega$ is defined by $\omega(v_i)=n_i$ and $n=n_1+\cdots+n_k$. As an example, we obtain a formula for the number of weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of three simplices.

algebraic topologycategory theorygroup theoryK-theory and homology

Audience: researchers in the topic

( video )

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Bilkent Topology Seminar

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