Hatcher, Algebraic Topology, Chapter 0
19. Show that the space obtained from by attaching 2-cells along any collection of circles in is homotopy equivalent to the wedge of 2-cells.
Let denote the space consisting of with circles in (that is, is a circle on the surface of for ), and let denote the space with 2-cells , , glued along the . Note that each as an embedded subspace in is contractible to a point despite the fact that 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.
It follows, then, that there exists a homotopy which contracts continuously all the to points on . It follows that the image is then a copy of with other copies of attached by the points to its surface, . Moreover, because there clearly exists a homotopy which fixes the “base” while sending all the points to the North Pole , say, the resulting space is then a wedge sum of copies of , that is, 2-cells joined by the common 0-cell .
In particular, then, the composition of and (with appropriately adjusted indices) is a homotopy connecting the given space to the wedge of 2-cells. Hence, the result.
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.