Graduate Thesis Or Dissertation
 

Splittings of skeletal homotopy modules

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/t148fk935

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  • This thesis is devoted to determining structure results on a group relative to a subgroup, using information about the kernel of the boundary map of associated free resolutions. If Y is a CW-complex with homotopy type K(G,1) then for n ≥ 2 the nth skeletal homotopy module, hn(Y ) = πn(Y (n)), is a kernel of the nth boundary homomorphism of a free resolution of Z by free ZG-modules. By passing to the universal cover and using cellular homology, analogous descriptions in dimensions zero and one are available. Let (Y,X) be a pair of connected, aspherical CW complexes of type K(G,1) and type K(S,1) respectively. If the map on fundamental groups induced by the topological inclusion is injective, then S can be seen as a subgroup of G and the induced skeletal homotopy module for X, ZG ⊗S hn(X), naturally injects into the nth skeletal homotopy module of Y . We define three conditions on this injection of the induced module. When it is split injective over ZS we say SumZS(G,S) holds, when it is split over ZG we say SumZG(G,S) holds, and when it is split injective with ZG-projective cokernel we say PSumZG(G, S) holds. When PSumZG(G,S) holds, a theorem of Serre [Hue79] implies that every finite subgroup of G is determined by S. When SumZG(G,S) holds, a theorem of Howie and Schneebeli [HS81] implies that the intersection of S with its conjugates is torsion free. When SumZS(G,S) holds, results of Bogley and Dyer [BD93] are generalized in this thesis to show that either S is self-normalizing in G or S has cohomological dimension less than or equal to n+1. We also study the relationships amongst these conditions. Our main result along these lines is that each of SumZS(G,S), SumZG(G,S) and PSumZG(G,S) imply the same at dimension n + 1 and hence all higher dimensions. Meanwhile we provide an example to show that not even PSumZG (G, S ) implies SumZS (G, S ). Moreover, clearly n−1 PSumZG(G,S) implies SumZG(G,S) which implies SumZS(G,S), but we provide examples to show neither converse holds in general. We apply these splitting results to cyclically presented groups on n generators. We show that if SumZG(G, S) holds for the semi-direct product of a cyclically presented group n on n generators with a cyclic group of order n, then the shift automorphism has order n. Using work of [BP92] we provide a family of cyclically presented groups whose shift automorphism has order n and apply a theorem of [CRS05] to determine that these groups cannot be the fundamental group of any hyperbolic 3-orbifold of finite volume.
  • Keywords: projective resolution over a group ring, relative asphericity, group structure, kernel, cohomology, malnormal, augmentation ideal, relation module
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