Classification of fiber sequences with a prescribed holonomy action

Complex Vasquez 不变量

Complex Vasquez invariant

该研究定义了 Vasquez 不变量的复模拟，将有限群视为紧平坦 Kähler 流形的完整群，以推广原不变量在平坦流形上的性质。

Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps with prescribed singular fibers

We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.

Fibering flat manifolds of diagonal type and their fundamental groups

An $n$-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group $G$ is diagonal. An $n$-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of $G$ as the least integer $n_d(G)$ such that every flat manifold of diagonal type with holonomy $G$ fibers over a flat manifold of dimension at most $n_d(G)$ with flat torus fibers. Using a combinatorial description of Bieberbach groups of diagonal type, we give both upper and lower bounds for this invariant. We s…

Obstructions for constructing equivariant fibrations

Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ is a $G$-equivariant fibration $p:E \rightarrow B$ where $p^{-1}(b)$ is $G_b$-homotopy equivalent to $F_{G_b}$ for each $b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\…

Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions

We classify fibrations of abstract $3$-regular GKM graphs over $2$-regular ones, and show that all fiberwise signed fibrations of this type are realized as the projectivization of equivariant complex rank $2$ vector bundles over quasitoric $4$-folds or $S^4$. We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension $6$ with prescribed one-skeleton, in particular with prescribed number of isolated fixed points.