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Full-Text Articles in Physical Sciences and Mathematics
Matching And Independence Complexes Related To Small Grids, Benjamin Braun, Wesley K. Hough
Matching And Independence Complexes Related To Small Grids, Benjamin Braun, Wesley K. Hough
Mathematics Faculty Publications
The topology of the matching complex for the 2 x n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(Δmn) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(Δmn). Furthermore, we determine the Euler characteristic of Ind(Δmn) and prove that several homology groups of Ind(Δmn) are non-zero.
Analytic And Topological Combinatorics Of Partition Posets And Permutations, Jiyoon Jung
Analytic And Topological Combinatorics Of Partition Posets And Permutations, Jiyoon Jung
Theses and Dissertations--Mathematics
In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent …
Algorithms For Upper Bounds Of Low Dimensional Group Homology, Joshua D. Roberts
Algorithms For Upper Bounds Of Low Dimensional Group Homology, Joshua D. Roberts
University of Kentucky Doctoral Dissertations
A motivational problem for group homology is a conjecture of Quillen that states, as reformulated by Anton, that the second homology of the general linear group over R = Z[1/p; ζp], for p an odd prime, is isomorphic to the second homology of the group of units of R, where the homology calculations are over the field of order p. By considering the group extension spectral sequence applied to the short exact sequence 1 → SL2 → GL2 → GL1 → 1 we show that the calculation of the homology …