abstract: One of the important aspect of hyperplane (or subspace) arrangement is to understand how far the topology of the complement of a union of hyperplanes (subspaces) is determined by the combinatorics of the arrangement. In recent years a lot of techniques from homotopy theory, notably the homotopy colimits have been used to discover some connections between combinatorics and topology. Homotopy colimit is an important idea originating in homotopy theory, that was developed by Quillen, Bousﬁeld, Kan and others. It has not only reached remarkable extension and depth but it has also proved to be a versatile tool in a lot of other areas of mathematics. The aim of my ﬁrst talk is to motivate and explain the construction of homotopy colimits and how it is used to understand the homotopy type of the arrangement complement. I will do it using examples and pictures avoiding technical jargon from category theory. In my second talk I will describe how the Bousﬁled-Kan spectral sequence is used to compute the (co)homology of the complement. If time permits I will also discuss the fundamental group of a homotopy colimit. These talks are based on the works of Welker-Ziegler-Zivaljevic 1999, Delucci 2006 and Dror Farjoun 2004 and some classical material on model categories.
Deshpande - Homotopy techniques 2