#### Date of Submission

9-28-2010

#### Date of Award

9-28-2011

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Mukherjee, Goutam (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The notion of cohomology with local coefficients for topological spaces arose with the work of Steenrod [Ste43, Ste99], in connection with the problem of extending sections of a fibration. This cohomology is built on the notion of fundamental groupoid of the space and can be described by the invariant cochain subcomplex of the cochain complex of the universal cover under the action of the fundamental group of the space. This later description is due to Eilenberg [Eil47]. Cohomology with local coefficients finds applications in many other situations.We focus on one such application of this cohomology which is due to S. Gitler [Git63], where he has constructed Steenrod reduced power operations in cohomology with local coefficients. The study of cohomology operations has been one of the important areas of research in algebraic topology for a long time. They have been extensively used to compute obstructions [Ste47], to study of homotopy type of complexes [Tho56] and to show essentiality of maps of spheres [BS53]. Some of the basic operations are the reduced powers of Steenrod [Ste53b, Ste53a]. These operations are defined for cohomology with coefficients, in a fixed cyclic group of prime order p â‰ 2. The main idea of Gitlerâ€™s construction is to lift power operations in the invariant cochain subcomplex of the universal cover and reproduce the operations in cohomology with local coefficients via Eilenbergâ€™s description of the cohomology with local coefficients, where the relevant local coefficients are obtained by a fixed action of the fundamental group of the space on a fixed cyclic group of prime order pâ‰ 2.Among many important roles played by Eilenberg-MacLane complexes, a significant one is its role in classifying cohomology. A simplicial version of this classification states that for any abelian group A and natural number q, the qth Eilenberg-MacLane simplicial set K(A, q) represents the q th cohomology group functor with coefficients in A, in the sense that for every simplicial set X, there is a bijective correspondence [Dus75]Hq (X; A) â‰… [X, K(A, q)].These classification results have been generalized for cohomology with local coefficients in [Hir79], [GJ99], [BFGM03], where generalized Eilenberg-MacLane complexes play the role of classifying spaces. A construction of a generalized Eilenberg-MacLane complex LÏ€X(L, q) is obtained in [BFGM03] as a homotopy colimit by using the method of Bousfield and Kan [BK72], where Ï€X is the fundamental groupoid of X and L is a local coefficient system on X. The complex LÏ€X(L, q) appears as the total space of a Kan fibration LÏ€X(L, q) âˆ’â†’ N(Ï€X), where N(Ï€X) denotes the nerve of the category Ï€X. The fibration may be interpreted as an object of the slice category S/N(Ï€X), where S denotes the category of simplicial sets. There is a canonical map Î· : X â†’ N(Ï€X) and the classification theorem states that the cohomology classes in the q th cohomology with local coefficients of a Kan complex X correspond bijectively to the vertical homotopy classes of liftings of Î·. The proof of course depends on the usual closed model structure for the category of simplicial sets.

#### Control Number

ISILib-TH303

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Sen, Debashis Dr., "Simplicial Bredon-Illman Cohomology with Local Coefficients." (2011). *Doctoral Theses*. 318.

https://digitalcommons.isical.ac.in/doctoral-theses/318

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843393