# Question about proof in Van Kampen's theorem; Hatcher

In the proof of Van Kampen's theorem in Hatcher's book, which is theorem 1.20 on p43 , we read in its proof on p45 (see here: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf)

If anybody wants, I can make a screenshot and post it here.

We may assume the $$s$$-partition subdivides the partitions giving the products $$f_1* \dots * f_k$$ and $$f_1'* \dots * f_l'$$.

What exactly does this line mean? What partitions giving the products $$f_1*\dots *f_k$$ is the author talking about? And why can we assume this?

The concatenation $$f_1 \cdot \ldots \cdot f_k$$ is a map $$I \to X$$ obtained by dividing the interval into $$k$$ subintervals and doing $$f_i$$ on the $$i$$th subinterval. You can take these subintervals to be $$[0, 1/k], [1/k, 2/k], \ldots, [(k-1)/k, 1]$$, or you can take the product two loops at a time; for instance, $$f_1 \cdot (f_2 \cdot (f_3 \cdot f_4)))$$ would use the partition $$[0, 1/2]$$, $$[1/2, 3/4]$$, $$[3/4, 7/8]$$, $$[7/8, 1]$$. This is the partition giving the product $$f_1 \cdot \ldots \cdot f_k$$ that Hatcher refers to (and similarly for $$f_1' \cdot \ldots \cdot f_l'$$).

When Hatcher says that the $$s$$-partition on page 45 can be assumed to subdivide the partitions giving the products, he means that the partition points (e.g. the 0, 1/2, 3/4, 7/8, and 1 in my example) each appear as one of the values $$s_0, s_1, \ldots s_m$$. We can assume this because if our $$s$$-partition doesn't contain these points, we simply refine the $$s$$-partition by inserting them; any refinement of the $$s$$-partition will still have the desired property that the rectangles $$R_{ij}$$ each map into a single $$A_{\alpha}$$.

• Thanks! I'll look into it!
– user661541
Commented Nov 27, 2019 at 22:52

I believe he means the following:

By the "partitions giving the products" I think he's means the how the product $$f_1\cdot \dots \cdot f_k$$ partitions $$I$$ into $$k$$ segments $$0 < \frac{1}{2^{k-1}} < \frac{1}{2^{k-2}} < \dots < \frac{1}{2} < 1$$. (Recall that in Lemma 1.15 he proves that loop $$f$$ in $$A$$ can be decomposed up to homotopy into a product $$f_1\cdot \dots \cdot f_k$$ where each $$f_i$$ is a loop contained entirely in some $$A_{\alpha_i}$$.)

Hatcher then considers a homotopy between two factorizations $$H\colon f_1\cdot \dots \cdot f_k \sim f_1'\cdot \dots \cdot f_l'$$ and shows how to make two partitions $$0= s_0 < s_1 < \dots < s_m = 1$$ (the "$$s$$-partition") and $$0= t_0 < t_1 < \dots < t_n = 1$$ (the $$t$$-partition) of $$I$$ with the property that for each $$i, j$$ we have $$H([s_i, s_{i+1}]\times [t_j, t_{j+1}])\subset A_{\alpha_{i, j}}$$ for some $$\alpha_{i, j}$$. If we add more values to our $$s$$-partition it will not change this property, and so in particular we can add the values from the partitions given by the factorizations $$f_1 \cdot \dots \cdot f_k$$ and $$f_1'\cdot \dots \cdot f_l'$$. I believe this is what he means by "We may assume..."

• Thanks! I'll look into it!
– user661541
Commented Nov 27, 2019 at 22:52