## Computation of cohomology operations for finite groups

Al-Baydli, Daher

Al-Baydli, Daher

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##### Repository DOI

##### Publication Date

2018-08-31

##### Type

Thesis

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##### 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

##### Publisher

NUI Galway

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##### Rights

Attribution-NonCommercial-NoDerivs 3.0 Ireland