Sergey Antonyan

Free and based-free involutions.

By a space we mean a separable metric space X. An involution is a continuous map f:X\to X such that f\circ f=Id_X. An involution is called free if it has no fixed points, i.e., f(x)\ne x, \forall x\in X. An involution is called based-free if it has a unique fixed-point.

The main goal of this talk is to provide a brief summary of recent results on free and based free involutions, published in the papers [1] and [2].

For Hilbert space \ell_2 we denote by \sigma the standard based-free involution \sigma: \ell_2\to \ell_2 given by the formula \sigma(x)= -x. Let B denote the unit ball in \ell_2, that is, B := \{x\in \ell_2 \mid \| x\| \le 1\}.

We prove that for every space (X, \tau) with a based-free involution, the equivariant maps (X, \tau)\to (B, \sigma) separate points and closed sets in X. That is, given a closet set A\subset X and a point x\in X\setminus A, there exists an equivariant map f: (X, \tau)\to (B, \sigma) such that f(x)\notin \overline{f(A)}. Here, the equivariantness of f means that it commutes with the given involutions, i.e., f(\tau (x))=\sigma (f(x)) for all x\in X.

This is applied to show that (\ell_2, \sigma) is universal in the sense that for each space with a based-free involution \tau, there exists an equivariant topological embedding (X, \tau)\hookrightarrow (\ell_2, \sigma). Similarly, we prove that the unit sphere \mathbb S=\{x\in \ell_2\mid \Vert x\Vert =1\}, endowed with the standard free involution x\mapsto -x, is universal for all spaces X with a free involution \tau:X\to X. Another universal space with a free involution is the punctured Hilbert cube [-1, 1]^\infty\setminus \{0\} endowed with the natural free involution x\mapsto -x.

We also investigate equivariant free and based-free compactifications and will present two new characterizations of based-free compactifications. It turns out that \mathbb S has no free compactification, while \ell_2 has no based-free compactification. We will explain why the countable product of real lines \mathbb R^\infty, endowed with the standard involution x\mapsto -x, is not universal for all based-free involutions. At the same time we will show that \mathbb R^\infty is universal for those based-free involutions which admit a based-free compactification.

This is a joint work with Jan van Mill (University of Amsterdam, The Netherlands) and James E. West (Cornell University, USA)

References

[1] S. Antonyan, J. van Mill and J. West, Based-free Involutions, Topology Appl. (2025), to appear.

[2] J. van Mill and J. West, Involutions of \ell_2 and s with unique fixed points, Trans. Amer. Math. Soc. 373 , no. 10 (2020), 7327—7346.