Enhanced cohomology and obstruction theory

Author

Marek Alfred Galecki

Date of Award

8-1993

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Lawrence S. Husch

Committee Members

Michall D. Vose, G. Samuel Jordan, Morwen Thinthethnrcinte

Abstract

For any n ≥ 4 and a finite regular CW-complex K we define an abelian group called the enhanced n-th cohomology group of K, denoted EHⁿ(K). These groups have (contravariant) functorial properties; in fact, if ƒ ≃ g:K→L then ƒ^* = g⁸:EHⁿ(L)→EHⁿ(K). Thus they are invariant on the homotopy type. There is an exact sequence (sequence in PDF) The enhanced cohomology groups give a well-defined obstruction to extend maps to some n-2-connected spaces from a subcomplex A of K to K^[n+1]∪A. We thus have the primary and secondary obstructions encompassed in a single, well-defined step. We also have a well-defined obstruction to embedding an n-dimensional simplicial complex in R^2n-1 The classical Steenrod paper (Ann. of Math., 48, 290-320) that introduced the squares produces a 2-step approach to solving the extension problem mentioned above, without realizing the existence of EH^.

Recommended Citation

Galecki, Marek Alfred, "Enhanced cohomology and obstruction theory. " PhD diss., University of Tennessee, 1993.
https://trace.tennessee.edu/utk_graddiss/10676