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487 \draw[green!50!brown] (\x,-2) -- (\x,2); |
487 \draw[green!50!brown] (\x,-2) -- (\x,2); |
488 } |
488 } |
489 \end{scope} |
489 \end{scope} |
490 \end{tikzpicture} |
490 \end{tikzpicture} |
491 $$ |
491 $$ |
492 \caption{Five examples of unions of pinched products}\label{pinched_prod_unions} |
492 \caption{Six examples of unions of pinched products}\label{pinched_prod_unions} |
493 \end{figure} |
493 \end{figure} |
494 |
494 |
495 Note that $\bd X$ has a (possibly trivial) subdivision according to |
495 Note that $\bd X$ has a (possibly trivial) subdivision according to |
496 the dimension of $\pi\inv(x)$, $x\in \bd X$. |
496 the dimension of $\pi\inv(x)$, $x\in \bd X$. |
497 Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision. |
497 Let $\cC(X)\trans{}$ denote the morphisms which are splittable along this subdivision. |
1406 Then there exist |
1406 Then there exist |
1407 \begin{itemize} |
1407 \begin{itemize} |
1408 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1408 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1409 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1409 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1410 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1410 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1411 such that $b_i$ and $b_{i+1}$both map to (glue up to) $a_i$. |
1411 such that $b_i$ and $b_{i+1}$ both map to (glue up to) $a_i$. |
1412 \end{itemize} |
1412 \end{itemize} |
1413 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
1413 In other words, we have a zig-zag of equivalences starting at $a$ and ending at $\hat{a}$. |
1414 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
1414 The idea of the proof is to produce a similar zig-zag where everything antirefines to the same |
1415 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
1415 disjoint union of balls, and then invoke Axiom \ref{nca-assoc} which ensures associativity. |
1416 |
1416 |