Karesi sıfır üst üçgensel matrislerüzerinde bir sanı ve Carlsson'ın mertebesanısı

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 differential graded module M over the polynomial ring A in r variables: If the homology of M is nontrivial and fi nite dimensional over the ground field, then N := dim_A M is at least 2^r. In this thesis, we state a stronger conjecture concerning varieties of square-zero upper triangular N x 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.