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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?

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2 Answers 2

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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 $A_{\alpha}$.

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  • $\begingroup$ Thanks! I'll look into it! $\endgroup$
    – user661541
    Commented Nov 27, 2019 at 22:52
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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..."

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

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