Some bijections cannot be smoothed

It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.

It is easy to see that, for equivariant sets with equality atoms X and Y, a finitely supported function f:X\to Y may exist even if an equivariant function from X to Y does not. For example, let X be a singleton and Y the set of atoms \mathbb{A}; one can map the unique element to an atom, but one needs to choose the atom first.

It is less clear what happens if the finitely supported function f is a bijection: does an equivariant bijection exist as well?

The answer is negative. For example, consider

  • X = \mathbb{A}\times \mathbb{N} + \{*\}
  • Y = \mathbb{A}\times\mathbb{N},

choose an atom a\in\mathbb{A} and define a function f:X\to Y by:

  • f(*) = (a,0)
  • f(a,n) = (a,n+1)
  • f(b,n) = (b,n) for b\neq a.

It is easy to see that f is a bijection supported by \{a\}. However, no equivariant bijection (or even an equivariant function) from X to Y exists, since X contains an element with empty support and Y does not.

Interestingly though, if X and Y are orbit-finite, then finitely supported bijections can be smoothed into equivariant ones. This was proved by Joanna, who in the near future will have written a  post about it.

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