Melissa Zhang, Boston College

Symmetries in topological spaces and homology-type invariants

 Abstract: Topologists often encounter spaces with interesting symmetries. By analyzing the symmetries of an object through the regularities of its algebraic invariants, we are able to learn more about the object and its relationship with smaller, less complex objects. For example, by using the right tools, we can easily see that for a topological space X equipped with a cyclic action, the rank of the singular homology of X is at least that of the fixed point set. 

In low-dimensional topology, knots and links are ubiquitous and far-reaching in their associations. One particular interesting algebraic invariant of links is Khovanov homology, a combinatorial homology theory whose graded Euler characteristic is the Jones polynomial. In this talk, we consider links exhibiting 2-fold symmetry and prove a rank inequality for a variant of Khovanov homology.