University of Illinois Urbana-Champaign

Simplicial Complexes and a Partial Classification of Almost Completely Decomposable Torsion Free Abelian Groups

Perez-Segui, Maria Luisa

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https://hdl.handle.net/2142/71238

Description

Title
Simplicial Complexes and a Partial Classification of Almost Completely Decomposable Torsion Free Abelian Groups
Author(s)
Perez-Segui, Maria Luisa
Issue Date
1985
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • The thesis treats the problem of how different two quasi-isomorphic groups can be.
  • We first attack the problem in a topological way. We construct a functor (kappa) from the category of pairs (G,U), where G is a topological group and U is an open symmetric neighborhood of O in G, into the category of simplicial complexes. We give all torsion free groups the discrete topology and denote by Tf the category thus obtained. The image under (kappa) of the Pontryagin dual of an inclusion in Tf is a regular simplicial covering map of connected complexes. The lifting criterion available in the theory of covering spaces is then used to get the following theorem: If L = N (CRPLUS) P, and for every y (ELEM) P (FDIAG) O and x (ELEM) N there is f: P (--->) N such that f(y) = x, then M decomposes.
  • Secondly we treat the problem algebraically in a constructive way. We restrict ourselves to the class of almost completely decomposable groups. We fix L = L(,1) (CRPLUS) . . . (CRPLUS) L(,n) with rank L(,i) = 1 for all i, inside a -vector space V of dimension n. We examine all subgroups M of V containing L such that (VBAR)M/L(VBAR) < (INFIN) and each L(,i )is pure in M. In the case when M/L is cyclic we define a normalized expression of a -basis of V. This expression is then used to classify up to isomorphism all M's as above in the case when M/L is cyclic and L has a totally disconnected type graph. The classification involves congruence relations of the coefficients of the normalized expression, and the primes dividing the L(,i)'s. As a corollary we obtain a complete classification of almost completely decomposable groups of rank 2.
Graduation Semester
1985
Type of Resource
text
Permalink
http://hdl.handle.net/2142/71238

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