342 and there are labels $c_i$ at the labeled points outside the blob. |
342 and there are labels $c_i$ at the labeled points outside the blob. |
343 We know that |
343 We know that |
344 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$ |
344 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$ |
345 and so |
345 and so |
346 \begin{align*} |
346 \begin{align*} |
347 \ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\ |
347 \pi\left(\ev(\bdy y)\right) & = \pi\left(\sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j}\right) \\ |
348 & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
348 & = \pi\left(\sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k\right) \\ |
349 & = 0 |
349 & = 0 |
350 \end{align*} |
350 \end{align*} |
351 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. |
351 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. |
352 |
352 |
353 The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly |
353 The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly |