# Hatcher 0.19

Hatcher, Algebraic Topology, Chapter 0

19. Show that the space obtained from $S^2$ by attaching $n$ 2-cells along any collection of $n$ circles in $S^2$ is homotopy equivalent to the wedge of $n+1$ 2-cells.

Proof.
Let $X_0$ denote the space consisting of $S^2$ with $n$ circles $C_1,\ldots,C_n$ in $S^2$ (that is, $C_i$ is a circle on the surface of $S^2$ for $i=1,\ldots,n$), and let $X_1$ denote the space $X_0$ with $n$ 2-cells $e^2_i$, $i=1,\ldots,n$, glued along the $C_i$. Note that each $C_i$ as an embedded subspace in $S^2$ is contractible to a point despite the fact that $S^1$ by itself is not; in his proof to a previous exercise, Tarun Chitra gives a nice visual showing this fact which is pasted below as Figure 1[1].

Figure 1

It follows, then, that there exists a homotopy $f_t$ which contracts continuously all the $C_i$ to points $x_i$ on $S^2$. It follows that the image $X=f_1(X_1)$ is then a copy of $S^2$ with $n$ other copies of $S^2$ attached by the points $x_i$ to its surface, $i=1,\ldots,n$. Moreover, because there clearly exists a homotopy $g_t$ which fixes the “base” $S^2$ while sending all the points $x_i$ to the North Pole $N$, say, the resulting space $g_1(X)$ is then a wedge sum of $n+1$ copies of $S^2$, that is, $n+1$ 2-cells joined by the common 0-cell $N$.

In particular, then, the composition of $f_t$ and $g_t$ (with appropriately adjusted indices) is a homotopy connecting the given space $X_0$ to the wedge $g_1(X)$ of $n+1$ 2-cells. Hence, the result.   $\square$

1. Note that the image in Figure 1 is a screenshot of the previously-linked document by Tarun Chitra and was borrowed without permission. As such, all rights to and credit for this image are solely his.