Open Collections

Description

The $TC$-generating function associated to a space $X$ is the formal power series $\mathcal{F}_X(x) = \sum_{r=1}^\infty TC_{r+1}(X)\,x^r.$ For many examples $X$, it is known that $\mathcal{F}_X(x)= \frac{P_X(x)}{(1-x)^2},$ where $P_X(x)$ is a polynomial with $P_X(1)=cat(X)$. Is this true in general I shall discuss recent developments concerning this question, including observing that the answer is related to $X$ satisfying logarithmicity of LS-category. Also, in the examples mentioned above, it is always the case that $P_X(x)$ has degree less than or equal to $2$. Is this true in general I shall discuss this question in the context of right-angled Artin (RAA) groups and along the way see how RAA groups yield some interesting byproducts for the study of $TC$.