Homotopy theory in toric topology

J. Grbić ; S. Theriault

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 71 (2016) no. 2, pp. 185-251

Résumé

In toric topology one associates with each simplicial complex $K$ on $m$ vertices two key spaces, the Davis–Januszkiewicz space $DJ_{K}$ and the moment-angle complex $\mathscr{Z}_{K}$, which are related by a homotopy fibration $\mathscr{Z}_{K}\xrightarrow{\widetilde{w}}DJ_K\to \prod_{i=1}^{m}\mathbb{C}P^{\infty}$. A great deal of work has been done to study the properties of $DJ_{K}$ and $\mathscr{Z}_{K}$, their generalizations to polyhedral products, and applications to algebra, combinatorics, and geometry. Chap. 1 surveys some of the main results in the homotopy theory of these spaces. Chap. 2 breaks new ground by initiating a study of the map $\widetilde{w}$. It is shown that, for a certain family of simplicial complexes $K$, the map $\widetilde{w}$ is a sum of higher and iterated Whitehead products. Bibliography: 49 titles.

Détail

Keywords: Davis–Januszkiewicz space, polyhedral product, higher Whitehead product, higher Samelson product.
Mots-clés : moment-angle complex, homotopy type