REU 2021

An REU project aimed at studying discrete models of equivariant topological complexity.

Course information


In my group, we will be exploring the topological complexity of equivariant finite spaces. Topological complexity is an invariant of topological spaces which measures, in some sense, how hard it is to plan paths through a given space. More formally, the idea is that, given a space \(X\) and points \(x,y\in X\), we want to choose a path from \(x\) to \(y\). We can think of such a path as a continuous function \(\alpha:[0,1]\to X\) which sends \(0\) to \(x\) and \(1\) to \(y\). If we want to choose a path for every pair of points, we need to define a function

\[X\times X\to \operatorname{Map}([0,1],X)\]

which takes each pair of points, and sends it to a path between these two points. Such a function is called a motion planning for \(X\).

Ideally, we’d be able to define a continuous motion planing, so that if \(x\) is close to \(a\) in \(X\), and \(y\) is close to \(b\) in \(X\), the the path we choose from \(x\) to \(y\) is close to the path we choose from \(a\) to \(b\). Unfortunately, this is very rarely possible. The topological complexity of \(X\) is a natural number \(\operatorname{TC}(X)\) which measures ‘how discontinuous a motion planning for \(X\) must be’. In particular, if \(\operatorname{TC}(X)=1\), then \(X\) admits a continuous motion planning.

Topological complexity for finite spaces has been studied fairly extensively by Tanaka. In his paper, Tanaka relates the topological complexity of finite spaces to a number of discrete invariants, as well as to the topological complexity of certain simplicial complexes (which are just topological spaces glued together out of generalized triangles).

Our aim will be to explore the topological complexity of finite spaces equipped with a continuous action of a finite group \(G\). In this setting, we will consider equivariant topological complexity as defined by Colman and Grant. By considering finite groups acting on finite spaces, we can put ourselves in a position to study a number of interesting feature, but which is still tractible to direct computation.