We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the ...

Title of Dissertation: Turaev Torsion of 3-Manifolds with Boundary Christopher B. Truman, Doctor of Philosophy, 2006 Dissertation directed by: Professor James A. Schafer Department of Mathematics We study the Turaev torsion of 3-manifolds with boundary; specifically how certain “leading order” terms of the torsion are related to cohomology operations. Chapter 1 consists mainly of definitions an...

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

We obtain generalizations of some results of Turaev from [Tur02]. Turaev’s results relate leading order terms of the Turaev torsion of closed, oriented, connected 3–manifolds to certain “determinants” derived from cohomology operations such as the alternate trilinear form on the first cohomology group given by cup product. These determinants unfortunately do not generalize directly to compact, ...