583 |
583 |
584 \subsection{The blob complex} |
584 \subsection{The blob complex} |
585 \label{sec:blob-definition} |
585 \label{sec:blob-definition} |
586 |
586 |
587 Let $X$ be an $n$-manifold. |
587 Let $X$ be an $n$-manifold. |
588 Assume a fixed system of fields. |
588 Assume a fixed system of fields and local relations. |
589 In this section we will usually suppress boundary conditions on $X$ from the notation |
589 In this section we will usually suppress boundary conditions on $X$ from the notation |
590 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
590 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
591 |
591 |
592 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
592 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
593 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
593 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
594 $\overline{X \setmin Y}$. |
594 $\overline{X \setmin Y}$. |
595 |
595 |
596 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
596 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
597 |
597 |
598 Define $\bc_0(X) = \lf(X)$. |
598 Define $\bc_0(X) = \lf(X)$. |
599 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
599 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
600 We'll omit this sort of detail in the rest of this section.) |
600 We'll omit this sort of detail in the rest of this section.) |
601 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
601 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
602 |
602 |
603 $\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
603 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
604 More specifically, define a 1-blob diagram to consist of |
604 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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605 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
605 \begin{itemize} |
606 \begin{itemize} |
606 \item An embedded closed ball (``blob") $B \sub X$. |
607 \item An embedded closed ball (``blob") $B \sub X$. |
607 %\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so. |
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608 %(See note in previous subsection.)} |
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609 %\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$. |
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610 \item A field $r \in \cC(X \setmin B; c)$ |
608 \item A field $r \in \cC(X \setmin B; c)$ |
611 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
609 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
612 \item A local relation field $u \in U(B; c)$ |
610 \item A local relation field $u \in U(B; c)$ |
613 (same $c$ as previous bullet). |
611 (same $c$ as previous bullet). |
614 \end{itemize} |
612 \end{itemize} |
615 %(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
613 In order to get the linear structure correct, we (officially) define |
616 %so we will omit $c$ from the notation.) |
614 \[ |
617 Define $\bc_1(X)$ to be the space of all finite linear combinations of |
615 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
618 1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
616 \] |
619 also the label ($u$ above) of the blob. |
617 The first direct sum is indexed by all blobs $B\subset X$, and the second |
620 \nn{maybe spell this out in more detail} |
618 by all boundary conditions $c \in \cC(\bd B)$. |
621 (See xxxx above.) |
619 Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
622 \nn{maybe restate this in terms of direct sums of tensor products.} |
620 |
623 |
621 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
624 There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
622 \[ |
625 combination of fields on $X$ obtained by gluing $r$ to $u$. |
623 (B, u, r) \mapsto u\bullet r, |
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624 \] |
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625 where $u\bullet r$ denotes the linear |
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626 combination of fields on $X$ obtained by gluing $u$ to $r$. |
626 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
627 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
627 just erasing the blob from the picture |
628 just erasing the blob from the picture |
628 (but keeping the blob label $u$). |
629 (but keeping the blob label $u$). |
629 |
630 |
630 Note that the skein space $A(X)$ |
631 Note that the skein space $A(X)$ |
631 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
632 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
632 |
633 |
633 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
634 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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635 local relations encoded in $\bc_1(X)$. |
634 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
636 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
635 2-blob diagrams (defined below), modulo the usual linear label relations. |
637 2-blob diagrams, of which there are two types, disjoint and nested. |
636 \nn{and also modulo blob reordering relations?} |
638 |
637 |
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638 \nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
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639 |
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640 There are two types of 2-blob diagram: disjoint and nested. |
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641 A disjoint 2-blob diagram consists of |
639 A disjoint 2-blob diagram consists of |
642 \begin{itemize} |
640 \begin{itemize} |
643 \item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$. |
641 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
644 %\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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645 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
642 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
646 (where $c_i \in \cC(\bd B_i)$). |
643 (where $c_i \in \cC(\bd B_i)$). |
647 \item Local relation fields $u_i \in U(B_i; c_i)$. |
644 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
648 \end{itemize} |
645 \end{itemize} |
649 Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$. |
646 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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647 reversing the order of the blobs changes the sign. |
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648 Define $\bd(B_0, B_1, u_0, u_1, r) = |
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649 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
650 In other words, the boundary of a disjoint 2-blob diagram |
650 In other words, the boundary of a disjoint 2-blob diagram |
651 is the sum (with alternating signs) |
651 is the sum (with alternating signs) |
652 of the two ways of erasing one of the blobs. |
652 of the two ways of erasing one of the blobs. |
653 It's easy to check that $\bd^2 = 0$. |
653 It's easy to check that $\bd^2 = 0$. |
654 |
654 |
655 A nested 2-blob diagram consists of |
655 A nested 2-blob diagram consists of |
656 \begin{itemize} |
656 \begin{itemize} |
657 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
657 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
658 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
658 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
659 (for some $c_0 \in \cC(\bd B_0)$). |
659 (for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
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660 \item A local relation field $u_0 \in U(B_0; c_0)$. |
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661 \end{itemize} |
660 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
662 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
661 (for some $c_1 \in \cC(B_1)$) and |
663 (for some $c_1 \in \cC(B_1)$) and |
662 $r' \in \cC(X \setmin B_1; c_1)$. |
664 $r' \in \cC(X \setmin B_1; c_1)$. |
663 \item A local relation field $u_0 \in U(B_0; c_0)$. |
665 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
664 \end{itemize} |
666 Note that the requirement that |
665 Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |
667 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
666 Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$. |
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667 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
668 As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
668 sum of the two ways of erasing one of the blobs. |
669 sum of the two ways of erasing one of the blobs. |
669 If we erase the inner blob, the outer blob inherits the label $r_1u_0$. |
670 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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671 It is again easy to check that $\bd^2 = 0$. |
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672 |
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673 \nn{should draw figures for 1, 2 and $k$-blob diagrams} |
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674 |
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675 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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676 (officially) |
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677 \begin{eqnarray*} |
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678 \bc_2(X) & \deq & |
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679 \left( |
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680 \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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681 U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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682 \right) \\ |
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683 && \bigoplus \left( |
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684 \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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685 U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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686 \right) . |
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687 \end{eqnarray*} |
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688 The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
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689 but we didn't feel like introducing a notation for that. |
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690 For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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691 (rather than a new, linearly independent 2-blob diagram). |
670 |
692 |
671 Now for the general case. |
693 Now for the general case. |
672 A $k$-blob diagram consists of |
694 A $k$-blob diagram consists of |
673 \begin{itemize} |
695 \begin{itemize} |
674 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
696 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
675 For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or |
697 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
676 $B_i \sub B_j$ or $B_j \sub B_i$. |
698 $B_i \sub B_j$ or $B_j \sub B_i$. |
677 (The case $B_i = B_j$ is allowed. |
699 (The case $B_i = B_j$ is allowed. |
678 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
700 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
679 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
701 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
680 %\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
702 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
681 %(These are implied by the data in the next bullets, so we usually |
703 (These are implied by the data in the next bullets, so we usually |
682 %suppress them from the notation.) |
704 suppress them from the notation.) |
683 %$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
705 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
684 %if the latter space is not empty. |
706 if the latter space is not empty. |
685 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
707 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
686 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$. |
708 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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709 is determined by the $c_i$'s. |
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710 $r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
687 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
711 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
688 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
712 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
689 If $B_i = B_j$ then $u_i = u_j$. |
713 If $B_i = B_j$ then $u_i = u_j$. |
690 \end{itemize} |
714 \end{itemize} |
691 |
715 |
692 We define $\bc_k(X)$ to be the vector space of all finite linear combinations |
716 If two blob diagrams $D_1$ and $D_2$ |
693 of $k$-blob diagrams, modulo the linear label relations and |
717 differ only by a reordering of the blobs, then we identify |
694 blob reordering relations defined in the remainder of this paragraph. |
718 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
695 Let $x$ be a blob diagram with one undetermined $n$-morphism label. |
719 |
696 The unlabeled entity is either a blob or a 0-cell outside of the twig blobs. |
720 $\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
697 Let $a$ and $b$ be two possible $n$-morphism labels for |
721 As before, the official definition is in terms of direct sums |
698 the unlabeled blob or 0-cell. |
722 of tensor products: |
699 Let $c = \lambda a + b$. |
723 \[ |
700 Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
724 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
701 Then we impose the relation |
725 \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
702 \eq{ |
726 \] |
703 x_c = \lambda x_a + x_b . |
727 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
704 } |
728 $\overline{c}$ runs over all boundary conditions, again as described above. |
705 \nn{should do this in terms of direct sums of tensor products} |
729 $j$ runs over all indices of twig blobs. |
706 Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
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707 of their blob labelings. |
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708 Then we impose the relation |
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709 \eq{ |
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710 x = \sign(\pi) x' . |
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711 } |
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712 |
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713 (Alert readers will have noticed that for $k=2$ our definition |
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714 of $\bc_k(X)$ is slightly different from the previous definition |
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715 of $\bc_2(X)$ --- we did not impose the reordering relations. |
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716 The general definition takes precedence; |
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717 the earlier definition was simplified for purposes of exposition.) |
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718 |
730 |
719 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
731 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
720 Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
732 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
721 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
733 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
722 If $B_j$ is not a twig blob, this involves only decrementing |
734 If $B_j$ is not a twig blob, this involves only decrementing |
723 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
735 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
724 If $B_j$ is a twig blob, we have to assign new local relation labels |
736 If $B_j$ is a twig blob, we have to assign new local relation labels |
725 if removing $B_j$ creates new twig blobs. |
737 if removing $B_j$ creates new twig blobs. |
726 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
738 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
727 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
739 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
728 Finally, define |
740 Finally, define |
729 \eq{ |
741 \eq{ |
730 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
742 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
731 } |
743 } |
732 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
744 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
733 Thus we have a chain complex. |
745 Thus we have a chain complex. |
734 |
746 |
735 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
747 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
736 |
748 |
737 |
749 \nn{?? remark about dendroidal sets} |
738 \nn{TO DO: |
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739 expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
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740 relations to Chas-Sullivan string stuff} |
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741 |
750 |
742 |
751 |
743 |
752 |
744 \section{Basic properties of the blob complex} |
753 \section{Basic properties of the blob complex} |
745 \label{sec:basic-properties} |
754 \label{sec:basic-properties} |