Special Year Learning Seminar

In this learning seminar talk, I will explain the local picture underlying the technique of defining integer-valued curve counts, known as the FOP perturbation (named after Fukaya-Ono and B. Parker). The problem is indeed very simple: given a finite group $G$ and two representations $V$ and $W$, despite that we may not have $G$-equivariant maps $f: V \to W$ which are transverse to $0\in W$, can one obtain an integral cycle from $f^{-1}(0)/G$? I will start with a few examples explaining why the naive consideration fails. Then I will introduce the idea of Fukaya-Ono and Parker, which specifies a class of maps when $V$ and $W$ are complex representations. Lastly, I will explain what "FOP transverse maps" are and why for such an $f: V \to W$, the isotropy free part of $f^{-1}(0)/G$ carries an integral cycle. Many examples will be provided. Except for basic facts about Whitney stratifications (which will be included in the talk), only elementary differential topology is needed to understand this talk. 

The talk is based on a joint work with Shaoyun Bai.