Hatcher 0.17

Hatcher, Algebraic Topology, Chapter 0

17. (a) Show that a mapping cylinder of every map f:S^1\to S^1 is a CW complex.

(b) Construct a 2-dimensional CW complex that contains both an annulus S^1\times I and a Möbius band as deformation retracts.

Proof. (a) Let f:S^1\to S^1 be arbitrary and consider the mapping cylinder S^1\sqcup_{f} S^1. In particular, the space X “looks like” a “hollow cylinder” S^1\times I attached to a copy of S^1 by the identification (x,1)\sim f(x) for all x in the “top copy” S^1\times\{0\} of S^1. The process to construct a CW structure on X is as follows:

  1. Begin with two points x_0=e^0_1 and f(x_0)=e^0_2.
  2. Construct the 1-skeleton X^1 of X by adding three 1-cells e^1_i, i=1,2,3:
    • Add two 1-cells e^1_1 and e^1_2 as disjoint circles for which e^0_i\in e^1_i, i=1,2.
    • Let e^1_3 be the 1-cell formed from the “graph” of f, drawn “around” the “cylindrical part” of S^1\times I.
  3. Attach a 2-cell e^2_1\cong D^2 by the attachment map that identifies \partial e^2_1\cong S^1 with the 1-skeleton X^1 formed above.

Figure 1 below shows a (very crude) illustration of what such a thing might look like. Note that the red curve in Figure 1 represents the “graph” of f, where the thickened red line represents the part of the graph “on the front” of the cylinder and the thinner line represents the part “on the back” of the cylinder.

Hatcher 0.17
Figure 1

(b) This problem follows a pattern similar to that in part (a). First, let M denote a Möbius band and let A denote an annulus. Here, note that both M and A retract onto their center circles S^1_M and S^1_A, respectively. Let f:M\to S^1 and g:A\to S^1 denote these retractions.

Let X_1 and X_2 be defined to be the mapping cylinders associated with f and g, respectively; that is,

X_1=S^1\sqcup_f M and X_2=S^1\sqcup_g A.

Note, in particular, that both spaces X_i, i=1,2, deformation retract onto their “range” subspaces as noted on page 2 of the text, i.e., X_1 deformation retracts onto M and X_2 deformation retracts onto A. Finally, one can form the desired space X by “gluing together” the spaces X_i, i=1,2, via the identity map on S^1. Visually, then, X looks like a paper towel roll with a Möbius band M glued to one end and an annulus A glued to the other, and by previous remarks, X deformation retracts onto the spaces M and A.   \square


2 thoughts on “Hatcher 0.17

  1. Hello again. I was now working on this problem now and ran into difficulties. I may be a little overly-pedantic here, but I’m unclear how you attached your 2-cell here. An example that I tried keeping in mind is if f: S^1 -> S^1 is a constant map then what you should end up with is a cone wedge S^1. Maybe this is what you meant. I wrote up a solution I thought you would be interested in as well: http://goo.gl/bBW86e

    Sorry if there is too much detail, but I was reviewing product CW complexes while I was writing everything out. (I don’t believe Hatcher really spends enough time explaining how the attaching maps for a product CW works–then in hindsight, it’s the only way it can possible make sense! But still… 🙂

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