Non-positive immersions

In a blog post about hyperbolicity of one-relator groups I mentioned the following property that a 2-dimensional complex \(X\) might have: \(X\) is said to have non-positive immersions if for every immersion of a finite, connected 2-complex \(Y\) into \(X\), we either have \(\chi(Y) \le 0\) or \(Y\) is contractible. For example, this property rules out the existence of immersed 2-spheres in \(X\).

(Note that sometimes variants of this notion are considered, like changing the contractibility conclusion for \(Y\) to the weaker \(\pi_1(Y) = 0\).)

A recent conjecture due to Wise was that every contractible 2-complex \(X\) actually has non-positive immersions. An affirmative answer to this would imply an affirmative answer to the long-standing open Whitehead asphericity conjecture (stating that every connected subcomplex of an aspherical 2-complex is again aspherical).

Now unfortunately, counter-examples to the above stated conjecture of Wise were constructed by Fisher (arXiv:2210.02304) and by Chemtov-Wise (arXiv:2210.01395).