Altküme kompleksinin genişleme sınıfı

he subset complex ∆(X) of a G-set X is defined as the simplicial complexwhose simplices are subsets of X. The oriented chain complex of ∆(X) givesa ZG-module extension of Z by Z (a copy of Z on which G acts via the sign|X|−1representation of RX) hence a class ζX ∈ ExtZG (Z, Z). This class was firstintroduced by Reiner and Webb in [21], and they asked in which cases it isnon-zero. In this thesis, we consider the mod 2 reduction ζ X of ζX and provethat ζ X is equal to the top Stiefel-Whitney class of the augmentation moduleIX = Ker{RX → R}. Then, we study the case where X = G/1, the transitive G-set with point stabilizer, and G is a 2-group. First, we show that ζ G/1 is zero whenG is non-abelian. Then, we calculate the top Stiefel-Whitney class explicitly forabelian 2-groups using the norm map and the tensor product formulas for Stiefel-Whitney classes, and decide when ζ G/1 is non-zero using the structure of groupcohomology for abelian groups. The main result of the thesis is that if G is2-group, then ζ G/1 is non-zero if and only if G ∼ (Z/2)n × Z/4.=Keywords: augmentation module, cohomology, vector bundles, Stiefel-Whitneyclasses, Euler class, extension class, spectral sequences.1