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Morse Numbers and Inequalities and Perfect Morse Functions for Integral Orbifold Borel Homology

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Sean Corrigan, Saint Louis University

What
  • PhD Oral Defense
When Wed, Jul 19, 2017
from 03:10 PM to 05:00 PM
Where Ritter Hall 323
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Abstract:
 

The critical points of an orbifold Morse function carry the data of a numeric index (akin to that of Morse functions on manifolds) as well as an orthogonal representation of a finite group, and we would like to use this data to relate orbifold Morse theory with the homology of orbifold classifying spaces. For effective orbifolds, these classifying spaces may be realized as Borel constructions, and with integer coefficients, the orbifold singularities will contribute torsion in infinitely many degrees of the associated Borel equivariant homology.

 

By using a notion of Morse functions on classifying spaces of compact Lie groups, we will define three types of Morse numbers for these Borel constructions, and two of these types will also depend upon the critical point data of an orbifold Morse function. In each case, we will demonstrate Morse inequalities in the spirit of Pitcher, where both the free and torsion generators in integral homology are compared with our Morse numbers. 

 

After describing a spectral sequence for orbifold Borel homology, which is afforded by an orbifold Morse function, we will shift focus to Morse functions that are perfect in the sense that the homology may be read immediately from the critical point data. We give a classification of all closed, orientable 2-orbifolds that admit such a function, and we will explain some necessary conditions for the existence of such a function on an orientable 3-orbifold. Time permitting, we will investigate low-dimensional examples where our Morse inequalities may be achieved as equalities, and we will discuss how this relates to the notion of perfect Morse functions. 

 
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