数理解析研究所講究録
第 1888 巻 2014 年 67-74

67

REGARDING 2-CHAINS WITH 1-SHELL BOUNDARIES IN ROSY THEORIESSUNYOUNG KIM(JOINT WORK WITH BYUNGHAN KIM AND JUNGUK LEE)YONSEI UNIVERSITY, KOREA

1. INTRODUCTION
In [6], Hrushovski developed connections between amalgamation properties and definable groupoids
for a stable theory: if a stable theory fails the 3-uniqueness property, then there exists a definablegroupoid. J. Goodrick and A. Kolesnikov constructed such groupoid in [5]. Furthermore J. Goodrick,associated to a family of amalgamation
B. Kim, and A. Kolesnikov developed homology groups
for strong types in stable theories. In particular, they showedfunctors and computed the groupthat if has n- $CA$ based on $A=$ acl $(A)$ for $n\geq 3$ , then $H_{n-2}=0$ for $p\in S(A)$ , thus $H_{1}(p)=0$ holds
for any simple $T.$
In this article, we work with amenable families of functors and corresponding homology groups
from [3],[4] to show $H_{1}(p)=0$ holds for a rosy , where is a Lascar type and classify all the possible
2-chains with a 1-shell boundary in a nontrivial amenable collection of functors.
This article is only intended to present a summary of the results from [7],[8] and we do not include
all the details of the proofs.
$T$$H_{n}$$H_{2}$

$T$

$T$

$p$

BASIC DEFINITIONSIn this section, we recall the basic definitions and facts which are established in [3],[4]. Throughout,is called downward closed if wheneversubsetdenotes some finite set of natural numbers.
$u\subseteq v\in X$ ,then $u\in X$ . Then as an ordered (by inclusion) set, $Xi$ a category. Before defining an
amenable family of functors, we introduce some notations. We fix a category . Given a functor
is the single inclusion map in$f:Xarrow C$ and $u\subseteq v\in X,$ $f_{v}^{u};=f(\iota_{u,v})\in Mor_{C}(f(u), f(v))$ where
$(u,
v)$
.
Mor
$A$

$s$

$X\subseteq \mathcal{P}(s)$$C$$\iota_{u,v}$(1) Let $X$ be a downward closed subset ofDefinition 1.1.
denotes the set$\mathcal{P}(s)$and let $t\in X$ . The symbol$X|_{t}$$\{u\in \mathcal{P}(s\backslash t)|t\cup u\in X\}\subseteq X.$, and $X$ as above, let : $Xarrow C$ be a functor. Then the localization of: $X|_{t}arrow C$ such thatfunctor(2) For

$s,$

$f$

$t$

$f$

at

$t$

is the$f|_{t}$$f|_{t}(v)=f(t\cup v)$ ,, for any $u\subseteq v\in X|t.$and(3) Letbe downward closed subsets, where and are finite sets ofandnatural numbers. Let : $Xarrow C$ and : $Yarrow C$ be functors. We say is a permutation of if
there is a bijection (not necessarily order-preserving) : $sarrow t$ such that $Y=\{\sigma(u) : u\in X\}$
. In this case we write $g=fo\sigma^{-1}.$
and for $v\subseteq w\in Y,$ $g(w)=f(\sigma^{-1}(w))$ andWe say that and are isomorphic if there are an order-preserving bijection : $sarrow t$
such that $Y=\{\tau(u) : u\in X\}$ and a family of morphisms $\{h_{u} : f(u)arrow g(\tau(u))|u\in X\}$ from
Mor $(C)$ such that for any $u\subseteq v\in X,$$(f|_{t})_{v}^{u}=f_{v\cup}^{u\cdot t}t$$X\subseteq \mathcal{P}(s)$$Y\subseteq \mathcal{P}(t)$

$f$

$s$

$t$

$f$

$g$

$g$

$\sigma$$(g)_{w}^{v}=f_{\sigma^{-1}(w)}^{\sigma^{-1}(v)}$$f$$\tau$$g$$h_{v}of_{v}^{u}=g_{\tau(v)}^{\mathcal{T}(u)}oh_{u}.$For example

$f$

and

$fo\sigma^{-1}$are isomorphic when$\sigma$is order-preserving.

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SUNYOUNG KIM

Remark 1.2. It easily follows that for a downward closed$X|_{t}=X\cap \mathcal{P}(s\backslash t)$

and in that case$X\subseteq \mathcal{P}(s)$and $t\in X$ , we have

iff $X=\cup\{\mathcal{P}(u)|t\subseteq u\in X\}$ ;$X|_{t}=\cup\{\mathcal{P}(u\backslash t)|t\subseteq u\in X\}.$: $Xarrow C$ for various non-empty
be a non-empty collection of functors
Definition 1.3. Let
downward-closed subsets $X\subseteq P(s)$ for all finite sets of natural numbers. We say that is amenable
if it satisfies all of the following properties.
: $Xarrow C$ is in , then every functor(1) (Closed under isomorphisms and permutations) If
or is isomorphic to is also in$g:Yarrow C$ which is either a permutation of
if and only if for(2) (Closed under restrictions and unions) Given a functor : $Xarrow C,$$f$$\mathcal{A}$$\mathcal{A}$

$s$

$f$

$\mathcal{A}$

$f$

$f$

$f$

every $u\in X$ , we have that$\mathcal{A}.$$f\in \mathcal{A}$$fr\mathcal{P}(u)\in \mathcal{A}.$(3) (Closed under localizations) Suppose that : $Xarrow C$ is in . Then for any $t\in X,$ $f|_{t}:X|_{t}arrow C$
is also in(4) (Extensions of localizations are localizations of extensions.) Let : $Xarrow C$ be in , and let(see Remark 1.2). Suppose that the localization
be such that
. Then there is a
: $X|_{t}arrow C$ has an extension $g:Zarrow C$ in for some$g_{0}|_{t}=g.$$Z_{0}=\{u\cup
v:u\inZ,v\subseteq
t\},$
, and
: $Z_{0}arrow C$ in such thatfunctor$f$$\mathcal{A}$$\mathcal{A}.$$\mathcal{A}$$f$$X|_{t}=X\cap \mathcal{P}(s\backslash t)$$t\in X\subseteq \mathcal{P}(s)$$(X|_{t}\subseteq)Z\subseteq \mathcal{P}(s\backslash t)$$\mathcal{A}$$f|_{t}$$f\subseteq g_{0}$$\mathcal{A}$$g_{0}$

Definition 1.4. Let $B\in$ Ob $(C)$ and suppose $f(\emptyset)=B$ . We say thatthat are over $B.$
denote the set of all functors$f$is over$B$

and we let$\mathcal{A}_{B}$$f\in \mathcal{A}$Let

be a non-empty amenable collection of functors mapping into the category$\mathcal{A}$$C.$(regular) -simplex in is a functor
Definition 1.5. Let $n\geq 0$ be a natural number.
with $|s|=n+1$ . The set is called the support of , or $supp(f)$ .
for some set. Then put
denote the collection of all regular -simplices in
Let$A$$S(\mathcal{A})$$n$$:= \bigcup_{B\in Ob(C)}S(\mathcal{A};B)$$f$:$\mathcal{P}(s)arrow C$

$f$

$s$

$s\subseteq\omega$$S_{n}(\mathcal{A};B)$

andLet
chains in$\mathcal{C}$$n$$S(\mathcal{A};B)$$\mathcal{A}_{B}$$:= \bigcup_{n}S_{n}(\mathcal{A};B)$.

; its elements are called ndenote the free abelian group generated by
$B$ .
$:=$
and
we
defineSimilarly,, or -chains over.
, where the ’s are distinct -simplices and the ’s are
If is an -chain in the form
nonzero integers, then we define the length of as $|c|=|n_{1}|+\cdots+|n_{k}|$ and the support of as theunion of the supports ofand the support of are always uniquely
Of course can be sometimes written as $(c+g)-g$ , but
computed in its standard form.
$S_{n}(\mathcal{A};B)$$C_{n}(\mathcal{A};B)$$C(\mathcal{A};B)$$n$$\mathcal{A}_{B}$$:= \bigcup_{n}C_{n}(\mathcal{A};B)$$C(\mathcal{A})$$\bigcup_{B\in Ob(C)}C(\mathcal{A};B)$$n$$f_{i}$$\sum_{1\leq i\leq k}n_{i}f_{i}$

$n$

$c$

$n_{i}$$c$$c$$f_{i}’ s.$

$|c|$

$c$

$c$

$f,$
. . to denote simplices and chains. Now we will define the bound. ,We useary operators and the homology groups.
$a,$ $b,$ $c,$ $\ldots,$$g,$ $h,$$\alpha,$$\beta,$:Definition 1.6. If $n\geq 1$ and $0\leq i\leq n$ , then the i-th boundary operator$s=\langles_{0}<\cdots<s_{n}\}$ , then
with
with
domain-simplexis defined so that if is a regular$\partial_{n}^{i}$$f$$C_{n}(\mathcal{A};B)arrow C_{n-1}(\mathcal{A};B)$$\mathcal{P}(s)$$n$$\partial_{n}^{i}(f)=f|\mathcal{P}(s\backslash \{s_{i}\})$and extended linearly to a group map on all ofIf $n\geq 1$ and $0\leq i\leq n$ , then the boundary map$C_{n}(\mathcal{A};B)$.

$\partial_{n}:C_{n}(\mathcal{A};B)arrow C_{n-1}(\mathcal{A};B)$$\partial_{n}(c)=\sum_{0\leq i\leq n}(-1)^{i}\partial_{n}^{i}(c)$We write$\partial^{i}$and$\partial$

for$\partial_{n}^{i}$and$\partial_{n}$, respectively, if$n$is defined by the rule.

is clear from context.

, and its elements are called $(n-)$ cycles. The
is denoted
Definition 1.7. The kernel ofare called $(n-)$ boundaries.
. The elements ofis denotedinimage of$\partial_{n}$$\partial_{n+1}$$C_{n}(\mathcal{A};B)$$Z_{n}(\mathcal{A};B)$$B_{n}(\mathcal{A};B)$$B_{n}(\mathcal{A};B)$

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REGARDING 2-CHAINS WITH 1-SHELL BOUNDARIES IN ROSY THEORIES

Since
to

$\partial_{n}0\partial_{n+1}=0,$$B_{n}(\mathcal{A};B)\subseteq Z_{n}(\mathcal{A};B)$and we can define simplicial homology groups relative$A_{B}.$

Definition 1.8. The nth (simplicial) homology group ofover$\mathcal{A}$$H_{n}(\mathcal{A};B) :=Z_{n}(\mathcal{A};B)/B_{n}(\mathcal{A};B)$

Remark/Definition 1.9. Let:
as follows: Let$supp(f_{i})=\{s_{i,0}<\cdots<s_{i,n}\}$ . Let a.

be a bijection. Then induces an automorphism, where each -simplex
with$:=\sigma_{i}(s_{i})=\{t_{0}<\cdots<t_{n}\}$and let. We define$\mathbb{N}arrow \mathbb{N}$$\sigma$$C_{n}(\mathcal{A}, B)arrow C_{n}(\mathcal{A}, B)$

is

$B$

$\sigma$$c= \sum_{i}n_{i}f_{i}\in C_{n}(\mathcal{A}, B)$$:=\sigma rs_{i}$$n$$f_{i}$:$\sigma^{*}$$s_{i}$$:=$$t_{i}$$\sigma^{*}(c):=\sum_{i}n_{i}(-1)^{|\sigma_{i}|}f_{i}\circ\sigma_{i}^{-1}$(see Definition 1.1 (3)) with$:=|\sigma_{i}’|$$|\sigma_{i}|$(so $=0$ or 1), where$\sigma_{i}’\in$

Sym $(n+1)$ such that for $j\leq n,$$\sigma_{i}(s_{i,j})=t_{\sigma_{i}’(j)}.$Now$\sigma^{*}$

commutes with , i.e.,$\partial$$\partial(\sigma^{*}(c))=\sigma^{*}(\partial(c))$

This can be inductively shown after verifying when.

is a transposition.

$\sigma$

Next we define the amalgamation properties. Notice that for $n=\{0, \ldots, n-1\}$ , we use$\mathcal{P}^{-}(n)$is$\mathcal{P}(n)\backslash \{n\}.$Definition 1.10. Let(1)$\mathcal{A}$$\mathcal{A}$be a non-empty amenable family of functors into a categoryhas -amalgamation if for any functor
such that$n$$g\supseteq f$$f$:$\mathcal{P}^{-}(n)arrow C,$$f\in \mathcal{A}$$C$

and let $n\geq 1.$, there is an $(n-1)$ -simplex$g\in \mathcal{A}.$has -complete amalgamation or n- $CA$ if has -amalgamation for every with $1\leq k\leq n.$(3)
has strong 2-amalgamation if whenever ::are simplices in and$fr\mathcal{P}(s\cap t)=gr\mathcal{P}(s\cap t)$ , then $f\cup g$ can be extended$h:\mathcal{P}(s\cupt)arrow C$ in
simplexto a(2)$\mathcal{A}$

$n$

$k$

$\mathcal{A}$$\mathcal{A}$

$f$

$k$

$\mathcal{P}(s)arrow C,$$g$$\mathcal{P}(t)arrow C$$\mathcal{A}$$\mathcal{A}.$Definition 1.11. An amenable family of functors
the strong 2-amalgamation property.

$\mathcal{A}$is called non-trivial if it is non-empty and satisfies

It easily follows that any non-trivial amenable family of functors contains an -simplex for each$n\geq 1$ . In the rest of
the paper, we shall only work with a non-trivial amenable familyof functors
into
$n$

$\mathcal{A}$$C.$

Definition 1.12. If $n\geq 1$ , an -shell is an -chain
$n$

$n$

$c$

of the form$\pm\sum_{0\leq i\leq n+1}(-1)^{i}f_{i},$where$f_{0},$$\ldots,$$f_{n+1}$

are -simplices such that whenever $0\leq i<j\leq n+1$ , we have$n$$\partial^{i}f_{j}=\partial^{j-1}f_{i}.$Remark/Definition 1.13. The boundary of an $(n+1)$ -simplex is an -shell, and the boundary of any-shell is . Note that has $(n+2)$ -amalgamation iff any -shell is a boundary of an $(n+1)$ -simplex.
For an $(n+1)$ -chain having an -shell boundary,is always an odd integer.
Now we introduce a weaker notion than 3-amalgamation:
has weak 3-amalgamation over $B$ if
$n$

$0$

$n$

$\mathcal{A}$

$n$

$c$

$n$

$|c|$

$\mathcal{A}$any 1-shell over$B$

is the boundary of a 2-chain over$B$of length$\leq 3.$The details of the following fact and corollaries can be found in [3],[4].
Fact 1.14. If$\mathcal{A}$

has $(n+1)-CA$ for some $n\geq 1$ , then$H_{n}(\mathcal{A};B)=$

Since

{

$[c]$

:

$c$

is an -shell over $B$ with support $n+2$ }.
$n$already has 2-amalgamation, we have thatboundary of some 2-chain over $B.$$\mathcal{A}$$H_{1}(\mathcal{A};B)$is trivial iff any 1-shell over $B$ is the

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SUNYOUNG KIM

Corollary 1.15. Assume that$p\in S(A)$ .
Corollary 1.16. If

$T$

$T$

has n- $CA$ over $A=$ acl $(A)$ for some $n\geq 3$ . Then $H_{n-2}(p)=0$ foris simple, then $H_{1}(p)=0$ for any complete type$p$

in $T.$

whose theory
From now on, we work with a large saturated model
the
small
sets ofonrelation
independence
is rosy if there is a ternary$iL$
here as thorn-independence.
independence properties [1],[2]. We take$\mathcal{M}=\mathcal{M}^{eq}$$\rangle L$

$T$

$T$

$\mathcal{M}$is rosy. Recall thatsatisfying the basic

denote the category of all small subsets
Now fix an algebraically closed set $B=$ acl $(B)$ , and let$(i.e.,$ fixing
pointwise) . For acontaining
and morphisms are elementary maps overof$u\subseteqv\inX$$:=f_{v}^{u}(f(u))\subseteqf(v)$. We now fix $p(x)\in S(B)$, we write
functor : $Xarrow C_{B}$ andwhere the tuple may possibly have an infinite arity.
$C_{B}$

$B$

$B$

$B$

$\mathcal{M}$$f_{v}^{u}(u)$

$f$

$x$

Definition 1.17.

$A$

closed independent functor in(1) $X$ is a downward-closed subset of
is of the form acl $(Cb)$ , where(2) For all non-empty $u\in X$ , we haveis a functor$p$

for some finite
is independent with$f(u)= ac1(B_{\cup}\bigcup_{i\in u}f_{u}^{(i\}}(\{i\}))$

and
Let

$\mathcal{A}(p)$Fact 1.18.

$\{f_{u}^{\{i\}}(\{i\})|i\in u\}$is independent over

$C=f_{(i\}}^{\emptyset}(\emptyset)$; and for $i\in s,$ $f(\{i\})$over $B.$;.

$f_{u}^{\emptyset}(\emptyset)$

denote all closed independent functors in$\mathcal{A}(p)$: $Xarrow C_{B}$ such that:$s\subseteq\omega;f(\emptyset)\supseteq B$$\mathcal{P}(s)$$b(\models p)$

$f$

$p.$

is a non-trivial amenable family of functors.

2. MAIN RESULT: $H_{1}(p)=0$ IN $RO$ SY THEORIES

We have $H_{1}(p)=0$ for any Lascar strong type which follows from the fact that Lascar distances
due to 3-amalgamation andare finite in rosy theories. Meanwhile the same holds for a simple$(so f(\emptyset)=B)$ , and $u=\{i_{0}< . . . <i_{k}\}\in X$ , we
: $Xarrow C_{B}$ in
Fact 1.14. For given$f(u)=[a_{0}, \ldots, a_{k}]$$f(u)=$, and acl $(a_{j}B)=f_{u}^{\{i_{j}\}}(\{i \})$ . Thusacl
, wherewrite
is independent over $B.$
$T$

$f$

$\mathcal{A}(p)_{B}$$(B, a_{0}\ldots a_{k})$$a_{j}\models p,$$\{a_{0}, \ldots, a_{k}\}$Theorem 2.1. If $B=M$ is a model, then$\mathcal{A}(p)$has weak 3-amalgamation over $M$ . Therefore$H_{1}(p)=0.$

be such that $a\equiv Bb$ . By the Lascar distance over $B$
Definition 2.2. Let a set $B$ and tuplesof and , denoted by $d_{B}(a, b)$ , we mean the smallest natural number such that there are tuples$a_{n}=b$ , where for each $a_{i}a_{i+1}(i<n)$ begins some $B$ -indiscernible sequence.
$a=a_{0},$
$a,$

$b$

$n$

$b$

$a$

$\ldots,$

Theorem 2.3. Suppose that the strong type$p$

is the Lascar (strong) type. Then $H_{1}(p)=0.$

to be . Given a 1-shell $f=a_{12}-a_{02}+a_{01}$ wheresuch that, we want to find a 2-chain
is a 1-simplex ineach:$a_{02}(\{2\})=[c]=a_{12}(\{2\})$,$a_{01}(\{1\})=[b]=a_{12}(\{1\})$
and
that
is
harm
to
assumeAgain therenoand $a_{01}(\{0\})$ $:=[d],$ $a_{02}(\{0\})$ $:=[d’]$ . By the extension axiom, we can assume that $\{b, c, d, d’\}$ is$d_{n}=d’$ , where $d_{i}d_{i+1}(i<n)$independent. Let
such that $d(d, d’)=n$ . So we have $d=d_{0},$sobegins an indiscernible sequence. Assume that

Proof For notational simplicity we may assume$a_{ij}$$\mathcal{P}(\{i,j\})arrow C_{B}$$B$$\emptyset$$S_{1}(\mathcal{A}(p))$$g$$d,$ $d’\models p$$\partial g=f.$$\ldots,$$b_{C\rangle}L_{dd’}d_{i}d_{n-1}$

Claim. There are $e_{i}\models p(i<n)$ such that$d_{i}d_{i+1}\backslash Le_{i}$and$b_{\mathcal{C}^{\backslash }}Ld_{0}\cdots d_{n}.$$e_{i}d_{i}\equiv e_{i}d_{i+1}.$

be an indiscernible sequence having a sufficiently large length.
. Since there are only boundedly manywith
Due to the extension axiom, we can choose

Proof of Claim. Let$I=\langle d_{i}d_{i+1}\cdots\rangle$$e_{i}’\equiv d_{i}$$e_{i}’\backslash LI$

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REGARDING 2-CHAINS WITH 1-SHELL BOUNDARIES IN ROSY THEORIES

types over , one can findmap that mapsto$e_{i}’$$f$$d_{i}d_{i+1}$$d_{j},$$d_{j’}(j<j’)$$d_{j}d_{j’}$. Thenwith. Due to the indiscernibility of , there is asatisfies
as desired.
$I$$e\’{i} dj\equiv e_{i}’d_{j’}$$e_{i}:=f(e_{i}’)$$e_{i}d_{i}\equiv e_{i}d_{i+1}$$\dashv$

Again by extension we suppose, so that each
is independent,
Also each $\{b, c, e_{n-1}\},$
is independent $(*)$ .
There is, whereare 2-simplices with support $u:=\{0,1,3\}$ such that $g_{0}^{+}(u)=$and(this follows from the above Claim); andextends. Hence
Similarly, we can find $:=g_{i}^{+}-g_{i}^{-}(0<i<n-1)$ , where eachis a 2-simplex with support
such that $g_{i}^{+}(u)=[d_{i}, b, e_{i}]$ and; and
Therefore we have$bc^{\lambda_{d_{1}d_{i+1}}}e_{i}$$\{b, d_{i+1}, e_{i}\}$$\{b, d_{i}, e_{i}\},$$\{c, d_{n}, e_{n-1}\}$$g_{0}$$[d_{0}, b, e_{0}]$$g_{0}^{+}$$:=g_{0}^{+}-g_{0}^{-}$$g_{0}^{+},$$g_{0}^{-}$$g_{0}^{-}(u)=[d_{1}, b, e_{0}];\partial^{0}g_{0}^{+}=\partial^{0}g_{0}^{-};\partial^{1}g_{0}^{+}=\partial^{1}g_{0}^{-}$$a_{01}$$(i.e., \partial^{2}g_{0}^{+}=a_{01})$$\partial g_{0}=a_{01}-\partial^{2}g_{0}^{-}.$$g_{i}^{+},$$g_{i}$$u$$g_{i}^{-}$$g_{i}^{-}(u)=[d_{i+1}, b, e_{i}];\partial^{0}g^{+}=\partial^{0}g^{-};\partial^{1}g^{+}=\partial^{1}g^{-}$$\partial^{2}g_{i}^{+}=\partial^{2}g_{i-1}^{-}.$$\partial(g_{0}+\cdots+g_{n-2})=a_{01}-\partial^{2}g_{n-2}^{-}.$$:=g_{n-1}^{+}-a_{023}+a_{123}$ , wherePut
is a 2-simplex with support $\{j, 2,3\}$ extending
such that $a_{023}(\{0,2,3\})=[d_{n}, c, e_{n-1}],$ $a_{123}(\{1,2,3\})=[b, c, e_{n-1}]$ . Alsois a 2-simplex with$g_{n-1}^{+}(\{0,1,3\})=[d_{n-1}, b, e_{n-1}]$ extending$(*)$. Moreover again by, we haveThus it follows$g_{n-1}$$a_{j23}$$a_{j2}$$g_{n-1}^{+}$$\partial^{1}g_{n-1}^{+}=\partial^{1}a_{023}.$$\partial^{2}g_{n-2}^{-}$$\partial g_{n-1}=\partial^{2}g_{n-1}^{+}-a_{02}+a_{12}=\partial^{2}g_{n-2}^{-}-a_{02}+a_{12}.$

Therefore $g:=g_{0}+\cdots+g_{n-1}$ satisfies$\partial g=f$

ae desired.

$\square$

3. CLASSIFICATIONIn this section, we classify 2-chains having 1-shell boundaries using two operations, the crossing
operation and the renaming support operation.
Remark/Definition 3.1. Suppose that an -chain

Assume that $j\in supp(c)\backslash supp(\partial(c))$ . In this case we say
Given $k\not\in s;=supp(c)$ , we let be a map sending to$j$$\sigma$

1.9,$\partial(\sigma^{*}(c))=\sigma^{*}(\partial(c))=\partial(c)$is given where each is an -simplex.
ahas vanishing support (in its boundary).
while fixing numbers in. Now as in$c= \sum_{i}n_{i}f_{i}$

$n$

$c$

$k$

$n$

$f_{i}$$s\backslash \{j\}$.

Definition 3.2.

(1) The crossing operation (or $CR$ -operation): Let and be 2-simplices with$supp(\beta)=\{i_{1}, i_{2}, i_{3}\}(i_{0}\neq i_{3})$ such that $\alpha r\mathcal{P}(\{i_{1}, i_{2}\})=\beta r$. Suppose that $\partial(\alpha+\epsilon\beta)(\epsilon=1 or -1)$ has no term (i.e., is cancelledout). Now by strong 2-amalgamation there is a 3-simplex with $supp(\delta)=\{i_{0}, i_{1}, i_{2}, i_{3}\}$ suchthat
and
. We takeand. Then it follows. Replacingbyis called the crossing operation. Hence from a 2-chain , if we obtain by the $CR$-operation(applied to two terms in c) thenand $|c’|\leq|c|.$(2) The renaming support operation (or $RS$-operation): This is basically what is described in 3.1
with $n=2$ . So let $c= \sum_{i}n_{i}f_{i}$ (-simplices) be a 2-chain having a vanishing support, say$j\in supp(c)\backslash supp(\partial(c))$ .
Let $k\not\in supp(c)$ . Then as in Remark/Definition 3.1, we can change
the support to and replace by someso that and have the same boundary.
This replacement of by is called the $RS$ -operation. In general, if is the result of by
applying the $RS$ -operation to a subsummand of , thenand $|d’|\leq|d|.$$\beta$$\alpha$$supp(\alpha)=\{i_{0}, i_{1}, i_{2}\},$$\mathcal{P}(\{i_{1}, i_{2}\})$$:=\gamma$$\gamma$$\gamma$$\delta$$\delta r\mathcal{P}(\{i_{0}, i_{1}, i_{2}\})=\alpha$$\beta’$$\alpha’$$\delta r\mathcal{P}(\{i_{1}, i_{2}, i_{3}\})=\beta$$:=\delta|\mathcal{P}(\{i_{1}, i_{2}, i_{3}\})$$:=\delta r\mathcal{P}(\{i_{0}, i_{2}, i_{3}\})$$\partial(\alpha+\epsilon\beta)=\partial(\alpha’+\epsilon\beta’)$$\alpha+\epsilon\beta$$\alpha’+\epsilon\beta’$

$c’$

$c$

$\partial(c)=\partial(c’)$$f_{i}2$$j$$k$$\mathcal{C}’;=\sigma^{*}(c)$

$c$

$c$

$c$

$c’$

$c’$

$d’$

$d$

$d$

$\partial(d)=\partial(d’)$

Remark/Definition 3.3.
(1) In general, the $CR$-operation is not symmetric. For example suppose that $c=f_{0}-f_{1}+f_{2}$ is given whereis a 2-simplex with $supp(f_{i})=\{0,1,2,3\}\backslash \{i\}$$f_{i}r\mathcal{P}(\{k,3\})=f_{j}r\mathcal{P}(\{k,3\})(\{i,j,k\}=\{0,1,2\})$ . Now aesume that by the $CR$
such that
operation, $f_{0}-f_{1}$ is replaced by $f_{4}-f_{2}$ where $supp(f_{4})=\{0,1,2\}$ andso that is$(f_{4}-f_{2})+f_{2}=f_{4}$replaced by. But is not obtained from using the $CR$-operation (unlessis redundantly written as $f_{4}-f_{2}+f_{2}$ ).
$f_{i}$-$\partial f_{4}=\partial c$$c$$f_{4}$$f_{4}$$c$

72
SUNYOUNG KIM(2) Now we say a 2-chain is proper if for any obtained from by finitely many applicationsof the $CR$ or $RS$ -operation (to subsummands), we have $|c|=|c’|$ . Among proper 2-chains,
now the $CR$ and $RS$ -operations are symmetric. Moreover clearly any 2-chain is reduced to aproper 2-chain by finitely many applications of the two operations.
We call proper 2-chains and are equivalent (written $c\sim c’$ ) if is obtained from byfinitely many applications of the $CR$ or $RS$ -operation to some subsummands. Hence if proper
and $|c|=|c’|.$and are equivalent then
$c’$

$c$

$c’$

$d$

$c’$

$c$

$c$

$c$

$c$

$\partial(c)=\partial(c’)$

Now we are ready to define the notions of two different types of 2-chain having a 1-shell boundary.

Definition 3.4. Let$\alpha$be a 2-chain having a 1-shell boundary.

(1) We call renameable type (or $RN$-type) if a subsummand of has a vanishing support. If
is not an $RN$ -type 2-chain $(so |supp(\alpha)|=3)$ we call non-renameable ( $NR$ -)type.
equivalent to there does not(2)
is said to be minimal if it is proper, and for any proper$\alpha$$\alpha$$\alpha$$\alpha$$\alpha’$$\alpha$$\alpha$exist a subsummand$\beta$of$\alpha’$such that$\partial(\beta)=0.$

For the notational simplicity, given a simplex
to denote. Also given a chain, wherewith $u=\{j_{0}, \ldots, j_{k}\}\subseteq supp(f_{i})$ , we write(in its unique form), we write
to denote$f_{i}^{j_{0},\ldots,j_{k}}$$c^{j_{0},\ldots,j_{k}}$$c= \sum_{i\in I}n_{i}f_{i}$$f_{i}r\mathcal{P}(u)$$\sum_{i\in J}n_{i}f_{i}$$f_{i}$$:=\{i\in I|supp(f_{i})=\langle j_{0}, \ldots,j_{k}\}\}.$$J$

For the rest of this section, we fix a 1-shell boundary $f_{12}-f_{02}+f_{01}$ with $supp(f_{jk})=\{j<k\}.$Definition 3.5. Let$2$-simplex) of$\alpha$$\alpha$be a 2-chain having the boundary $f_{12}-f_{02}+f_{01}.$

is called a chain-walk in$\alpha$from$f_{01}$$A$

subchain$\beta=\sum_{i=0}^{m}\epsilon_{i}b_{i}(b_{i}$to $-f_{02}$ if$k_{m+1}$ (not necessarily distinct) such that $k_{0}=1,$ $k_{m+1}=2,$(1) there are non-zero numbers
and for $i\leq m,$ $supp(b_{i})=\{k_{i}, k_{i+1},0\}$ ;; and(2) each(3) for $0\leq i<m,$$k_{0},$$\ldots,$$\epsilon_{i}\in\{1, -1\};(\partial\epsilon_{0}b_{0})^{0,1}=f_{01},$$(\partial\epsilon_{m}b_{m})^{0,2}=-f_{02}$$(\partial\epsilon_{i}b_{i})^{0,k_{i+1}}+(\partial\epsilon_{i+1}b_{i+1})^{0,k_{+1}}\cdot=0.$

Notice that such a representation is sensitive to its order, and a chain-walk can have distinct represubchain
sentations. Unless said otherwise a chain-walk is written in a form of a representation.
$A$of the chain-walkof a form$\beta$$\beta’$$:=m\acute{\sum_{i=j}}\epsilon_{i}b_{i}$for some $0\leq j<m’\leq m$ is called a section of$\beta.$$A$

chain-walk in is called maximal (in ) if it has the maximal possible length. We say is centeredat if (so every) maximal chain-walk in fromto $-f_{02}$ is, as a chain, equal to, and that is centered at 2, and so on are similarly defined.
Now a chain-walk in $\alpha from-f_{02}$ to$\beta$$0$$\alpha$$\alpha$$\alpha$$a$$f_{01}$$\alpha$$f_{12}$

Lemma 3.6. Let$\alpha$$\alpha.$$\alpha$be a 2-chain with the 1-shell boundary $f_{12}-f_{02}+f_{01}$ . Leta chain-walk in , say from$\alpha$$-f_{02}$to$f_{12}$. Assume there is a section$\beta’=\sum_{i=\acute{j}}^{m}\epsilon_{i}b_{i}$of$\beta=\sum_{i=0}^{m}\epsilon_{i}b_{i}$$\beta$besuch that for, either $k_{i}\neq k_{m’+i}$ for all $i\in\{j, \ldots, m’\}$ ; orfor all $i\in\{j+1, \ldots, m’+1\}.$$CR$-operation to , we obtain a 2-simplex with $supp(c)=$Then by finitely many applications of the$supp(b_{i})=\{2, k_{i}, k_{i+1}\}$$k_{i}\neq k_{j}$$\beta’$$\{2, k_{j}, k_{m’+1}\}$and$\epsilon=1$

or $-1$ so that$c$$\beta";=\sum_{i=0}^{j-1}\epsilon_{i}b_{i}+\epsilon c+\sum_{i>m}^{m},$$\epsilon_{i}b_{i}$is still a chain-walk from $-f_{02}$ to$f_{12}.$

Theorem 3.7. Let(1) Assume$f_{01}$$\alpha$$\alpha$be a minima12-chain with the boundary $f_{12}-f_{02}+f_{01}.$

is of $NR$-type. Then $|\alpha|=1$ or

to $-f_{02}$ is of the form

some $1\leq j\leq n-1.$$\sum_{i=0}^{2n}(-1)^{i}a_{i}$$|\alpha|\geq 5$. If$|\alpha|\geq 5$then any chain-walk inwhich is as a chain equal to$\alpha$such that$\alpha$from$f_{12}=a_{2j}^{1,2}$

for

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REGARDING 2-CHAINS WITH 1-SHELL BOUNDARIES IN ROSY THEORIES(2)$\alpha$is of $RN$ -type iff$\alpha$is equivalent to a 2-chain$\alpha’=a_{0}+\sum_{i=1}^{2n-1}\epsilon_{i}a_{i}+a_{2n}$$(n\geq 1)$

which is a chain-walk from $f_{01}to-f_{02}$ such that $supp(a_{2n})=\{0,1,2\}$ and$\partial^{0}a_{2n}=f_{12},$$\partial^{1}(a_{2n})=-f_{02}.$

Proof. (1) This is easy to check,(2) Here we give a brief sketch of the left to right.

Note that
Claim 1. There is$|\alpha|\geq 3.$$(\Rightarrow)$$\alpha_{1}\sim\alpha$centered at 2 such that $|supp(\alpha_{1})|>3.$

Claim 2. There is a 2-chain
such thathas a 1-simplex term (with the coefficient 1)
such that $supp(c)=\{0,1,2\}$ , and $f_{12}-f_{02}=\partial^{0}(c)-\partial^{1}(c)$ .
Then let us take a chain-walk from $f_{01}to-f_{02}$ in
terminating with . By repeatedly applying
the $CR$-operations to (while unchanged), we obtain a desired
centered at forming a
chain-walk from $f_{01}to-f_{02}$ . Then we reorder the representation of the chain-walk if necessary.
$\alpha_{2}\sim\alpha_{1}$$c$$\alpha_{2}$$\gamma$$\gamma$$c$$\alpha_{2}$$\alpha’\sim\alpha_{2}$

$c$

$0$

$\alpha’$$\square$

The following theorem is proved using the notions of directed graph theory which are not coveredin this note.
Theorem 3.8. Let be a minima12-chain having the 1-shell boundary $f_{12}-f_{02}+f_{01}$ . Thenequivalent to a 2-chain which is a chain-walk fromto $-f_{02}$ such that $supp(\alpha’)=\{0,1,2\}.$$\alpha$$\alpha$is$f_{01}$In the next section, we explore some of the consequences of this theorem.

4. ApPLICATION : MATRIX EXPRESSION
In this section, we introduce the notion of a matrix expression, which determines whether a given
minima12-chain having a 1-shell boundary is of $RN$ -type.
For the rest, we fix a minima12-chain of length $2n+1$ with the 1-shell boundary $f_{12}-f_{02}+f_{01},$
and $supp(\alpha)=\{0,1,2\}$ . For $\{0,1,2\}=\{i, j, k\},$denotes $f_{J^{k}}\prime(j<k)$ . Fix $I=\{0,1,2\}$ and$J=\{0, \ldots, n\}$ . Also, we write
to denote that a 2-simplex term
is in$\alpha$$f_{i}’$$\epsilon a\in\alpha$Definition 4.1. Let$\sum_{j=0}^{2n}(-1)^{j}a_{j}$$\epsilon a$

be a representation of the given. By a matrix expression of (the representation of)$-f_{1}’$$\alpha$$\alpha$$\alpha.$which is a chain-walk from$f_{2}’$

to, we mean a function $M$ : $I\cross Jarrow J$ suchthat(1) for each $i\in I,$ $Mr\{i\}\cross J$ : $(\{i\}\cross)Jarrow J$ is a permutation of ;(2) for each $i\in I,$ $M(i, 0)$ is an index such that(3) for each $i\in I,$$M(i,j)$ is an index such that$J$

$f_{i}’=\partial^{i}a_{2M(i,0);}$$j\in J\backslash \{O\},$$\partial^{i}a_{2j-1}=\partial^{i}a_{2M(i,j)}.$Interpret $M(i, j)$ as an entry
of a matrix in the $(i+1)$ th row and the $(j+1)$ th column, then$M=(m_{ij})_{I,J}$ is a $3\cross(n+1)$ matrix.
$m_{ij}$Notice that matrix expressions are determined according to the choices of pairs of terms whichcancel out each other.
Theorem 4.2. The following conditions are equivalent:(1)$\alpha$is of $RN$ -type.

(2) There is a matrix expression $M$ for a representation, and non-empty
under the function $M.$
$0\leq i_{0}<i_{1}\leq 2$$J_{0}\subseteq\{1, \ldots, n\},$$\alpha=\sum_{j=0}^{2n}(-1)^{j}a_{j}$such that for some$M(\{i_{0}\}\cross J_{0})=M(\{i_{1}\}\cross J_{0})$as image set

74
SUNYOUNG KIM

Proof.

$(\Rightarrow)$Let$\alpha_{1}$be a subchain of$\alpha=\sum_{j=0}^{2n}(-1)^{j}a_{j}$such that$\partial^{i}\alpha_{1}=0$

for$i\in I\backslash \{i_{*}\}$

and $|\alpha_{1}|=2m,$$:=\{j\in$where $i_{*}\in\{0,1,2\}$ is a vanishing support. Let
for eachand$J|-a_{2j-1}\in\alpha_{1}\}$ . Then here $M$ satisfies Definition 4.1 and $M(\{i_{0}\}\cross J_{0})=M(\{i_{1}\}\cross J_{0})$ , where, as desired.
Suppose that $M(\{i_{0}\}\cross J_{0})=M(\{i_{1}\}xJ_{0})$ , say , where
and $0\leq i_{0}<i_{1}\leq 2.$, a subsummand of . Then we have
, so
Let
has a
$i\in I\backslash \{i_{*}\}$$a_{2M(i,j)}\in\alpha_{1}$$j\in J_{0}$$\{i_{0}, i_{1}, i_{*}\}=I$$\alpha_{1}$$J_{0}\subseteq\{1, \ldots, n\}$$J_{1}$$(\Leftarrow)$$\partial^{i_{0}}\alpha_{1}=\partial^{i_{1}}\alpha_{1}=0$$\alpha$$:= \sum_{j\in J_{1}}a_{2j}+\sum_{j\in J_{0}}-a_{2j-1}$

vanishing support$i_{*}$$\alpha_{1}$, where $\{i_{0}, i_{1}, i_{*}\}=I.$$\square$

We end this note by stating some consequences of Theorem 4.2 which can be proved by using
permutations induced from matrix expressions.
For a matrix expression $M$ : $I\cross Jarrow J$ , there is a tripleof permutations of suchthat
is a map sending the $(i+1)th$ row to the $(k+1)th$ row, i.e., $\sigma_{ik}(m_{ij})=m_{kj}$ for $j\in J$ , and$0\leq i<k\leq 2$ . Notice that$(\sigma_{01}, \sigma_{12}, \sigma_{02})$$J$$\sigma_{ik}$$\sigma_{02}=\sigma_{12}\circ\sigma_{01}.$

Theorem 4.3. If$n$is odd, thenThen$\alpha$is always of $RN$ -type.

as in Definition 4.1, one of the following holds:;
for some $0<j_{0}<j_{1}\leq n$ andfor some $0\leq j_{0}<j_{1}\leq n$ andTheorem 4.4. Suppose that for(1)(2)$\alpha$$\alpha$$0\leq\ell\leq 2$$\partial^{\ell}a_{2j_{0}-1}=\partial^{\ell}a_{2j_{1}-1}$$\partial^{\ell}a_{2j_{0}}=\partial^{\ell}a_{2j_{1}}$$0\leq\ell\leq 2.$

is of $RN$ -type.

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3. J. Goodrick, B. Kim, and A. Kolesnikov, Amalgamation functors and homology groups in model theory, Preprint.
4. –, Homology groups of types in model theory and the computation of(p) , Journal of Symbolic Logic 78$H_{2}$(2013), 1086-1114.

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$S$$Z$