## Computation of cohomology operations for finite groups

##### View/Open

##### Date

2018-08-31##### Embargo Date

2022-08-31

##### Author

Al-Baydli, Daher

##### Metadata

Show full item record##### Usage

This item's downloads:

**0**(view details)

##### Abstract

The main result of this dissertation is the computation of all Steenrod squares on the
Mod 2 cohomology of all groups of order dividing 32 and all but 58 groups of order
64; partial information on Steenrod square is obtained for all but two groups of order
64. For groups of order 32 this thesis completes the partial results due to Rusin[21],
Thanh Tung Vo [25] and Guillot [12].
The thesis also demonstrates how the underlying techniques can be used to compute
the Stiefel-Whitney class of a certain real representations. Other contributions of the
dissertation are the following (in which denotes a finite group and denote -modules):
1. We devise and implement an algorithm for computing the induced cohomology
homomorphism (See Algorithm 3.1.1.)
2. We devise and implement an algorithm for computing the induced homology
homomorphism. (See Algorithm 3.3.1.)
3. We devise and implement an algorithm for calculating the connecting
cohomology homomorphism that inputs a surjective morphism of --modules
with kernel, and an integer and outputs the connecting cohomology
homomorphism. (See Algorithm 3.2.1).
4. We devise and implement an algorithm that inputs a surjective morphism of -
modules with kernel, and an integer and outputs the connecting homology
homomorphism.
(See Algorithm 3.4.1).
5. We compute the Bockstein homomorphism for finite -groups. (See Section
6.2.6).
6. We give a group cohomology construction and implementation of the cupproduct.
(See Section 4.1).
7. . We use the cup-i product to compute the Steenrod square on the classifying
space of a finite 2-groups. (See Section 4.2).
8. We devise and implement an algorithm for cohomology detection that inputs a
finite group, one or more maximal subgroups and an integer and outputs
information on the Steenrod squares

##### Collections

This item is available under the Attribution-NonCommercial-NoDerivs 3.0 Ireland. No item may be reproduced for commercial purposes. Please refer to the publisher's URL where this is made available, or to notes contained in the item itself. Other terms may apply.

The following license files are associated with this item: