535 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.04}{hochschild/0-chains}$; the base point |
535 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.04}{hochschild/0-chains}$; the base point |
536 in $S^1$ is labeled by $m$ and there are no other labeled points. |
536 in $S^1$ is labeled by $m$ and there are no other labeled points. |
537 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
537 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
538 as shown in Figure \ref{fig:hochschild-1-chains}. |
538 as shown in Figure \ref{fig:hochschild-1-chains}. |
539 |
539 |
540 \begin{figure}[ht] |
540 \begin{figure}[t] |
541 \begin{equation*} |
541 \begin{equation*} |
542 \mathfig{0.4}{hochschild/1-chains} |
542 \mathfig{0.4}{hochschild/1-chains} |
543 \end{equation*} |
543 \end{equation*} |
544 \begin{align*} |
544 \begin{align*} |
545 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
545 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
546 \end{align*} |
546 \end{align*} |
547 \caption{The image of $m \tensor a$ in the blob complex.} |
547 \caption{The image of $m \tensor a$ in the blob complex.} |
548 \label{fig:hochschild-1-chains} |
548 \label{fig:hochschild-1-chains} |
549 \end{figure} |
549 \end{figure} |
550 |
550 |
551 \begin{figure}[ht] |
551 \begin{figure}[t] |
552 \begin{equation*} |
552 \begin{equation*} |
553 \mathfig{0.6}{hochschild/2-chains-0} |
553 \mathfig{0.6}{hochschild/2-chains-0} |
554 \end{equation*} |
554 \end{equation*} |
555 \caption{The 0-chains in the image of $m \tensor a \tensor b$.} |
555 \caption{The 0-chains in the image of $m \tensor a \tensor b$.} |
556 \label{fig:hochschild-2-chains-0} |
556 \label{fig:hochschild-2-chains-0} |
557 \end{figure} |
557 \end{figure} |
558 \begin{figure}[ht] |
558 \begin{figure}[t] |
559 \begin{equation*} |
559 \begin{equation*} |
560 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
560 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
561 \end{equation*} |
561 \end{equation*} |
562 \caption{The 1- and 2-chains in the image of $m \tensor a \tensor b$. |
562 \caption{The 1- and 2-chains in the image of $m \tensor a \tensor b$. |
563 Only the supports of the blobs are shown, but see Figure \ref{fig:hochschild-example-2-cell} for an example of a $2$-cell label.} |
563 Only the supports of the blobs are shown, but see Figure \ref{fig:hochschild-example-2-cell} for an example of a $2$-cell label.} |
564 \label{fig:hochschild-2-chains-12} |
564 \label{fig:hochschild-2-chains-12} |
565 \end{figure} |
565 \end{figure} |
566 |
566 |
567 \begin{figure}[ht] |
567 \begin{figure}[t] |
568 \begin{equation*} |
568 \begin{equation*} |
569 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
569 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
570 \end{equation*} |
570 \end{equation*} |
571 \begin{align*} |
571 \begin{align*} |
572 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |
572 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |