# Extending a covering space of the 2-skeleton of a CW complex to the CW complex

Let $$Y$$ be a CW-complex and $$f: Y^2 \to Y$$ be the inclusion of its two skeleton. We define $$f^*Z := \{(y, z) \in Y^2 \times Z \ | \ y = f(y) = p(z)\}$$ and $$f^*p : f^*Z \to Y^2$$ the restriction of the projection $$Y^2 \times Z \to Y^2$$.

Question

Show that for every covering map $$q: W \to Y^2$$, there exists a covering $$p: Z \to Y$$ and a homeomorphism $$g: W \to f^*Z$$ such that $$q = (f^*p)\circ g$$.

Possible approach

A CW complex has a universal cover and $$\pi_1(Y^2) = \pi_1(Y)$$, so there is a bijection between coverings of $$Y^2$$ and $$Y$$ by the classification theorem of coverings. Take $$p$$ to be the covering corresponding to $$q$$. Also this is part b of a larger question, part a was to prove covering maps are stable under pullback.

• What is $Z$, and what is $p$? – Santana Afton Apr 5 at 11:43
• $Z$ is a topological space and $p$ is a covering map. They are not given, we must prove that they exist such that they satisfy the requirements. – Pel de Pinda Apr 5 at 13:41

## 1 Answer

In your question you describe the general construction of a pullback of a map $$p : Z \to Y$$ along a map $$f : X \to Y$$. Pullback diagrams are characterized by a universal property, so pullbacks are unique up to homeomorphism. Now it is easy to see that if $$f$$ is the inclusion of a subspace, then the restriction $$p_ X = p \mid_{p^{-1}(X)} : p^{-1}(X) \to X$$ and the inclusion $$f_X : p^{-1}(X) \to Z$$ build a pullback diagram with $$p,f$$. Hence instead of $$f^*p$$ we may take $$p_X$$ which is somewhat easier to visualize.

Your question therefore in fact means to find an extension of $$q : W \to Y^2$$ to a covering $$p : W' \to Y$$ (i.e. $$W \subset W'$$ and $$p \mid_W = q$$).

$$Y^3$$ is obtained from $$Y^2$$ by attaching $$3$$-cells, similarly we get $$Y^4, Y^5, \dots$$. Now each attaching map $$\phi : S^2 \to Y^2$$ lifts to $$\tilde{\phi} : S^2 \to W$$ because $$\pi_1(S^2) = 0$$. More precisely, for each $$w \in q^{-1}(\phi(*))$$ we get a unique lift $$\tilde{\phi}_w : S^2 \to W$$ such that $$\tilde{\phi}_w(*) = w$$. Use the collection of all these $$\tilde{\phi}_w$$ to attach $$3$$-cells to $$W$$. This yields a CW-complex $$W^3 \supset W$$ and an extension $$q^3 : W^3 \to Y^3$$ of $$q$$ (the open $$3$$-cell of $$W^3$$ belonging to $$\tilde{\phi}_w$$ is mapped homeomorphically onto the the open $$3$$-cell of $$Y^3$$ beloging to $$\phi$$). Preceeding skeletonwise we get the desired extension $$p : W' \to Y$$.