#### Date of Submission

2-28-1991

#### Date of Award

2-28-1992

#### 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, Amiya (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In this thesis we shall present an equivariant analogue of the steenrod cohomology with local coefficients, and use this to develop an obstruction therory for equivariant fibrations. The work is inspired by a remark of tom dieck [2] which says that a sensible translation of the classical obstruction theory to equivariant fibrations uses (meaning 'should use') equivariant cohomology with local coefficients.The equivariant singular cohomology of illman (9] is unsuitable for obstruction theory for equivariant sections of an equivariant fibration. The difficulty lies in connecting the obstruction cochains which arise from different fixed point sets, and the situation becomes no better even if we work with the Bredon cohomology [1]. Explicitly, if (B, A) is a G-complex pair, and p : E-B is a G-fibration with fibre F, and if o is an equivariant section over BnUA, then, for two (n+1)-cells a and o of B, the values c(a), (o) of the obstruction cochain c lie in different groups. In fact we do not have a coefficient system M such that both c(o) and cglo) belong to the same group M(G/Go), even in the case when Go =G as there is no canonical isomorphism between n(Fo) and n(F).The main problem of the thesis is to extend the local cohomology of Steenrod [18]. (20] to the category of G-spaces where G is a compact group, so that the resulting cohomology fits well into equivariant obstruction theory. We construct for a G-space X cohomology HA(X, M), where M is a suitable equivariant local coefficients system on X. The cohomology satisfies all the equivariant Eilenberg-Steenrod axioms. and can also be described in terms of the equivariant cellular structure when X is a G-CW-complex. Moreover, it reduces to the Steenrod cohomology with classical local coefficients system[20] when G is trivial, and to the equivariant singular cohomology with contravariant coefficient system of Illman (9] when M is simple in certain sense and X is G-path connected. The key idea behind the construction of our cohomology lies in generalizing the classical fundamental groupoid to the equivariant fundamental groupoid of a G-space. The equivariant fundamental groupoid reduces to the classical fundamental groupoid when the group is trivial, and is the basis of the definition of a local coefficients system on a G-space. The local nature of this construction reflects the distinetion of our cohomology theory with that of Illman. Whereas our coefficients system depends heavily on the space with which it is associated, the contravariant coefficient system of Illnan is independent of the space.We then build up an obstruction theory for equivariant section of G-fibration using our cohomology, when G is finite. This is accomplished by allowing the classical obstruction theory to dominate on each fixed point set X, and then piecing them together over the whole of X in a natural way. Note that if G is finite, then each XH inherits an ordinary CW structure from the equivariant CW structure of X (13). This is the basis of our obstruction theory. It seems that some kind of global description of the obstruction cochain, like that cosidered by Whitehead [20], is necessary to tackle the- general case when G is compact.

#### Control Number

ISILib-TH182

#### 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

Mukherjee, Goutam Dr., "Equivariant Cohomology with Local Coefficients." (1992). *Doctoral Theses*. 107.

https://digitalcommons.isical.ac.in/doctoral-theses/107

## 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:28842883