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Top ology on Graphs
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Zhi Lu Ё Institute of Mathematics, Fudan University, Shanghai.

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Osaka, 2006
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§1. Ob jective
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· Graphs = · · · = Geometric Ob jects

· Two basic problems
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- - - Under what condition, is a geometric ob ject a closed manifold? - - - Can any closed manifold be geometrically realizable by the above way?

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§2. Background · GKM theory--by Goresky, Kottwitz and MacPherson in 1998 (see [Invent. Math. 131, 25-83]).
GKM manifolds T
k

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M

2n

A unique regular GKM graphs

M

of valency n
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A GKM manifold is a T -manifold M with · |M T | < + · M having a T k -invariant almost complex structure · for p M T , the weights of the isotropy representation of T k on TpM being pairwise linearly independent.

k

2n

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§3. Coloring graphs and faces
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Let G = (Z2)k . Given a G-manifold M with |M G| < with properties as follows: a natural map : EM - Hom(G, Z2) e - such that

regular graph M

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A) for each p VM , (Ep) spans Hom(G, Z2) B) for each e = pq EM and (Ep), the number of times which and + (e) o ccur in (Ep) is the same as that in (Eq ).

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-- Abstract definition Let G = (Z2)k . We shall work on H (B G; Z2) = Z2[a1, ..., ak ] ( Hom(G, Z2)).

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H 1(B G; Z2) =

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n: a connected regular graph of valency n with n k and no loops. If there is a map : E - H 1(B (Z2)k ; Z2) - {0} s. t. (1) for each vertex p V, the image (Ep) spans H 1(B (Z2)k ; Z2), and (2) for each edge e = pq E, (x )
xEp -Ee y Eq -Ee
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(y )

mod (e),

then the pair (, ) is called a coloring graph of typ e (k , n).
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-- Examples
a3

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(, 1 ) is a coloring graph
a1 a2

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a1

a2 a3

1 : E - H 1 (B (Z2 )3 ; Z2 ) where H (B (Z2 )3 ; Z2 ) = Z2 [a1 , a2 , a3 ].

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a3

(, 2 ) is not a coloring graph
a1 a1 + a2

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a1

a2 a3

a1 a2 a1 (a1 + a2 ) mod a

3

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--Faces (, ): a coloring graph of type (k , n). : a connected -valent subgraph of where 0 n. If ( , | ) satisfies a) for any two vertices p1, p2 of , ((E | )p1 ) and ((E | )p2 ) span the same subspace of H 1(B G; Z2); b) for each edge e = pq E | , (x )
x(E | )p -(E | )e y (E | )q -(E | )e

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(y ) mod (e)
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then ( , | ) is an -face of (, ).
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Example

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a2 a1 a2 a3 a1

is a 2-face
a1 + a2

a1 + a2

a2 + a3

a1 + a3

a coloring graph (, )
a1 a3

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is not a 2-face
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a1 + a2

a2 + a3
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Assumption--Case: valency n of = rank k of G = (Z2) (, ): a coloring graph of type (n, n) with connected. F(,): the set of all faces of (, ). -- An application for the n-connectedness of a graph.

k
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Theorem (Whitney) A graph with at least n + 1 vertices is n-connected if and only if every subgraph of , obtained by omitting from any n - 1 or fewer vertices and the edges incident to them, is connected. Theorem (Z. Lu and M. Masuda). Suppose that (, ) is a Ё coloring graph of type (n, n) with connected. If the intersection of any two faces of dimension 2 in F(,) is either connected or empty, then is n-connected.

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Example
a a
2 1

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a a2 + a

3

a
3

3

a

2

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3

a1 + a

2

a1 + a

a2 + a3 a1 + a3 a1 + a a
1

2

a a
2

1

a a
2

1

a

3

a

3
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a1 + a

2

a1 + a2 a a
1

a1 + a

3

a1 + a a
1

3
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a2 + a
2

3

a2 + a
3

3

a
3

3

a1 + a2

a1 + a

a

3

a a2 + a
3

2

a1 + a

3

a1 + a2

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a2 + a

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§4. Geometric realization (, ):a coloring graph of type (n, n)=F(,)=|F(,)| Example 1.
four 2-faces:
a a a1 + a
2 1

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1

a

1

a1 + a a
3

a1 + a
3

2

a1 + a a
3

3

a1 + a

2

a1 + a

3

a a
2

2

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a2 + a

3

a

2

a

a2 + a
3

3
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(, ):a coloring graph of type (3, 3)

a2 + a

3

The geometric realization |F(,)| = S

2

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Example 2.
a1 a3 a1 a2 a3 a2 a1 a2 a3 a1 a3 a2 a3 a3

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(, ):a coloring graph of type (3, 3).
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a1 a2

The geometric realization |F(,)| = RP

2

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a2
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a1
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Generally, Fact. F(,) forms a simplicial poset of rank n with respect to reversed inclusion with (, ) as smal lest element. |F(,)| is a pseudo manifold. poset means partially ordered set A poset P is simplicial if it contains a smallest element ^ and for each a P the segment [^, a] is a boolean 0 0 algebra (i.e., the face poset of a simplex with empty set as the smallest element).

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P : a simplicial poset a simplicial cell complex KP in the following way: for each a = ^ in P , one 0 such that its face poset all obtained geometrical to the ordered relation in complex as desired. obtains a geometrical simplex is [^, a], and then one glues 0 simplices together according P , so that one can get a cell

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By |P | one denotes the underlying space of this cell complex, and one calls |P | the geometric realization of P .

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Basic problems: (I). Under what condition, is the geometric realization |F(,)| a closed topological manifold? (II). For any closed topological manifold M n, is there a coloring graph (, ) of type (n + 1, n + 1) such that M n |F(,)|?

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Basic problem (I) (, ): a coloring graph of type (n, n) with connected. The case n = 1: |F(,)| S
0

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The case n = 2: it is easy to see that for any coloring graph (, ) of type (2, 2), the geometric realization |F(,)| is always a circle. The case n = 3: Fact.|F(,)| is a closed surface S .

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Generally, if n > 3, the geometric realization |F(,)| is not a closed topological manifold. For example, see the following coloring graph (, ) of type (4, 4).
a
2

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a a a
3 1

3

a

1

a

2

a

4

a

4

a

4

a

4

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a a a
2 3

1

a a
1

2

a

3
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(|F(,)|) = 5 - 12 + 16 - 8 = 1 = 0 so |F(,)| is not a closed topological 3-manifold.

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The case n = 4. Write v = |V| and e = |E| so 2v = e. f : the number of all 2-faces in F(,) f3: the number of all 3-faces in F(,) Theorem. Let n = 4. |F(,)| is a closed connected topological 3-manifold f = f3 + v .

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Problem: for n > 4, to give a sufficient (and necessary) condition that |F(,)| is a closed connected topological manifold.

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Basic problem (I I) M n:n-dim closed connected topological manifold 1-dim case: M S . S1 is realizable by any coloring graph of type (2, 2).
1 1

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2-dim case: Prop. Any closed surface can be realized by some coloring graph of type (3, 3).

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3-dim case: Conjecture: Any closed 3-manifold M 3 is geometrical ly realizable by a coloring graph (, ) of type (4, 4), i.e., M 3 |F(,)|. 4-dim case: It is well known that there exist closed topological 4-manifolds that don't admit any triangulation. closed topological 4-manifolds that cannot be realized by any coloring graph of type (5, 5).

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Prop osition. Let M n be a closed manifold. If M n admits a simplicial cel l decomposition with at least n + 2 vertices, then M n can be geometrical ly realizable by a coloring graph.

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Restatement Prop osition. Suppose that is a 3-valent graph and is at least 2-connected. Then is planar if and only if admits a coloring of type (3, 3) such that |F(,)| S 2.

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