Equivalence of path-connected CW-complexes and CW-complexes with one 0-cell

Question

Proposition Any path-connected CW-complex is homotopy equivalent to a CW-complex with precisely one 0-cell.

Proof (Sketch) Let $X$ be a path-connected CW-complex, so $sk_1(X)$ is a connected graph. Let $\Gamma\subset sk_1(X)$ be a spanning tree. Since $\Gamma$ is contractible, the map $X\to X/\Gamma$ is a homotopy equivalence.

Question What kind of quotient space is $X/\Gamma$? How should I visualize it?

Answer 1

Let $X$ be a CW-complex and $A$ a subcomplex, that means $A$ is a union of cells $\{e_\alpha^n\mid\alpha\in J \}$ such that $\overline{e_\alpha^n}$ is also contained in $A$. Then $A$ is closed and the quotient map $q:X\to X/A$ is a closed surjection. Such maps preserve the $T_4$-property of $X$. In particular, $X/A$ is Hausdorff. We can then apply the following proposition which translates between the intrinsic and the constructive definition of a CW complex:

Given a Hausdorff space $X$ and a family of maps $\Phi_\alpha : D_\alpha^n \to X,$ then these maps are the characteristic maps of a CW complex structure on $X$ iff :
(i) Each $\Phi_\alpha$ restricts to an injection from int$D_\alpha^n$ onto its image, a cell $e_\alpha^n ⊂ X,$ and these cells are all disjoint and their union is $X.$
(ii) For each cell $e_\alpha^n$ , $\ \Phi_\alpha(∂D_\alpha^n)$ is contained in the union of a finite number of cells of dimension less than $n.$
(iii) A subset $C$ of $X$ is closed iff $\Phi_\alpha^{-1}(C)$ is closed in $D_\alpha^n$ for each cell $e_\alpha^n$.

As the new characteristic maps we take $\Psi_\beta=q\circ\Phi_\beta$ for each cell $e_\beta^n$ which is not in $A$, and a single map $D^0\to\{A\}$ as the new $0$-cell the set $A$ was shrunk to. These maps clearly satisfy condition (i). It is also easy to see that they satisfy (ii) since the image of $\partial D_\alpha^n$ can only be in fewer cells after collapsing $A$. Condition (iii) says that $X$ has the final topology of all $\Phi_\alpha$, but since $q$ is a quotient map, also $X/A$ will have the final topology of all $\Psi_\beta$.

So this gives us a natural cell complex structure on the quotient space.