#### Date of Submission

2-28-1994

#### Date of Award

2-28-1995

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Mukherjee, Amiya (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

This thesis has grown out of efforts to understand in an equivariant set up two old problems which were revived and developed extensively by Quillen, and also by others. The first problem concerns a de Rham type theorem over the field of rationals Q for a G-simplicial set where G is a finite group. The second problem deals with equivariant plus construction, classification of equivariant acyclic maps, and equivariant G-homotopy type of a G-space where G is a compact Lie group. More specifically, the content of the thesis is governed by the following two theorems, and our main objective is to look for their generalizations in suitable equivariant categories.Theorem A (Cartan [7)). If A is a cohomology theory over a commuta- tive ring R with 1, and A(K) is the associated differential algebra of a simplicial set K, then there is a natural isomorphism of graded R-modulesH(A(K)) â‰… H'(K; R(A)},where R(A) is an R-module functorially determined by A.Here a differential algebra means a differential graded algebra, which is a graded R-module âŠ•pâ‰¥0MP having a differential &: Mp âŸ¶ Mp+1 with 2 0, and a multiplication MPMpâŸ¶Mp+qsatisfying the Liebnitz rule &alpha(Î±Î²) = (sÎ±)Î²+ (-1)Pa(6Î²). A cohomology theory is a contravariant functor from the indexing category âˆ‡ for simplicial objects to the category DGA of differential algebras over R satisfying certain axioms (sce Chapter 6).Theorem B (Kan-Thurston (15). If X is a path connected CW-space, then there exists a group a with a perfect normal subgroup N such that X has the homotopy type of K(âŠ° 1)N+.Here K(âŠ° 1) is the space obtained from the Eilenberg-MacLane space K(âŠ°, 1) by applying the plus construction of Quillen [25] with respect to N.Theorem A has its origin in the commutative cochain problem which was posed by Thom in 1957. A solution to this problem entails in constructing a contravariant functor A: TOP â€“ DGA so as to yield a de Rham type theorem which asserts that there is an isomorphismH(A(X)) â‰… H(X; R)for every topological space X, where the cohomology on the right is the singular cohomology. For example, the classical de Rham theorem provides a solution for the subcategory of smooth manifolds where A'(X) is the differential aigebra over the field of reals R of smooth differential forms on a manifold X. On the other hand, there does not exist a differential algebra over the integers z (the commutativity fails) and, as realized by Steenrod some 50 years ago, this accounts for the existence of cohomology operations, such as Steenrod squares, etc.In [24], Quillen solved the rational commutative cochain problem in an ab- stract setting and the solution obtained is rather complicated. Later Sullivan (27] gave another proof using his theory of minimal models and the de Rham complex A(K) of rational polynomial forms on a simplicial set K. An inde- pendent proof, which is based on an earlier proof by Thom in the real case, was given by Swan (28] when the coefficient ring R is a field of characteristic zero. Finally, Cartan (7] formulated the main ideas of Swan in the form of axioms for a cohomology theory, and proved Theorem A.

#### Control Number

ISILib-TH214

#### 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

Naolekar, Aniruddha C. Dr., "Some Problems in Equivariant Topology." (1995). *Doctoral Theses*. 97.

https://digitalcommons.isical.ac.in/doctoral-theses/97

## 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:28842873