A conjecture on square-zero upper triangular matrices and Carlsson's rank conjecture

Date

2018-09

Authors

Şentürk, Berrin

Editor(s)

Advisor

Ünlü, Özgün

Supervisor

Co-Advisor

Co-Supervisor

Instructor

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Abstract

A well-known conjecture states that if an elementary abelian p-group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. Carlsson gives an algebraic version of this conjecture by considering a di erential graded module M over the polynomial ring A in r variables: If the homology of M is nontrivial and nite dimensional over the ground eld, then N := dimAM is at least 2r. In this thesis, we state a stronger conjecture concerning varieties of square-zero upper triangular N N matrices with entries in A. By stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when N < 8 or r < 3. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture as well as novel results for N > 4 and r = 2.

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Book Title

Degree Discipline

Mathematics

Degree Level

Doctoral

Degree Name

Ph.D. (Doctor of Philosophy)

Citation

Published Version (Please cite this version)

Language

English

Type

Thesis