Four-term chain with three zero maps on second homotopy

Construct finite two-dimensional CW-complexes X₁ ⊂ X₂ ⊂ X₃ ⊂ X₄ such that π₂(X₁) is nonzero and each of the three inclusion-induced maps on π₂ is zero.

Background

The paper constructs a three-term chain K(𝒳) ⊂ K(𝒴) ⊂ K(𝒵) in which the first complex is non-aspherical and both inclusion-induced maps on π₂ are zero. It explains that an arbitrarily long chain of this type could potentially yield an aspherical union containing a non-aspherical subcomplex, thereby providing a counterexample to Whitehead’s asphericity question. The authors explicitly state that they do not know how to extend their construction to a four-term chain with three successive zero maps.

References

This result leads to the following natural problem: how long can a chain of inclusions of two-complexes be if the first complex is non-aspherical and every inclusion induces the zero map on \pi_2? In particular, at present the author does not know how to construct

X_1\subset X_2\subset X_3\subset X_4

with \pi_2(X_1)\ne0 and all three inclusion-induced maps on \pi_2 equal to zero.

— Two results on asphericity  (2608.20270 - Mikhailov, 20 Aug 2026) in Introduction