equal
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30 then $\Compat(D^\bullet_*)$ is $i$-connected. |
30 then $\Compat(D^\bullet_*)$ is $i$-connected. |
31 \end{thm} |
31 \end{thm} |
32 |
32 |
33 \begin{proof} |
33 \begin{proof} |
34 (Sketch) |
34 (Sketch) |
35 This is a standard result; see, for example, \nn{need citations: Spanier}. |
35 This is a standard result; see, for example, \cite[Chapter 4]{MR0210112}. |
36 |
36 |
37 We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. |
37 We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. |
38 Choose $f(x_{0j})\in D^{0j}_0$ for all $j$ |
38 Choose $f(x_{0j})\in D^{0j}_0$ for all $j$ |
39 (possible since $D^{0j}_0$ is non-empty). |
39 (possible since $D^{0j}_0$ is non-empty). |
40 Choose $f(x_{1j})\in D^{1j}_1$ such that $\bd f(x_{1j}) = f(\bd x_{1j})$ |
40 Choose $f(x_{1j})\in D^{1j}_1$ such that $\bd f(x_{1j}) = f(\bd x_{1j})$ |