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].

Hatcher 0.19
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.

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