Topology on Graphs Title Page Contents Zhi L u Institute of - PowerPoint PPT Presentation

Home Page Topology on Graphs Title Page Contents Zhi L¨ u ◭◭ ◮◮ Institute of Mathematics, Fudan University, Shanghai. ◭ ◮ Page 1 of 22 Osaka, 2006 Go Back Full Screen Close Quit

Home Page § 1. Objective Title Page Contents Graphs = ⇒ · · · = ⇒ Geometric Objects • ◭◭ ◮◮ ◭ ◮ • Two basic problems Page 2 of 22 − − − Under what condition, is a geometric object a closed manifold? Go Back − − − Can any closed manifold be geometrically realiz- Full Screen able by the above way? Close Quit

Home Page § 2. Background Title Page • GKM theory—by Goresky, Kottwitz and MacPherson in 1998 (see [Invent. Math. 131 , 25-83]). Contents GKM manifolds ◭◭ ◮◮ T k � M 2 n ◭ ◮ A unique regular Γ M of valency n Page 3 of 22 GKM graphs A GKM manifold is a T k -manifold M 2 n with Go Back  • | M T | < + ∞ Full Screen   • M having a T k -invariant almost complex structure   • for p ∈ M T , the weights of the isotropy representation Close   of T k on T p M being pairwise linearly independent.   Quit

Home Page § 3 . Coloring graphs and faces Title Page Let G = ( Z 2 ) k . Given a G -manifold M with | M G | < ∞ � regular graph Γ M Contents with properties as follows: ◭◭ ◮◮ ∃ a natural map α : E Γ M − → Hom( G, Z 2 ) ◭ ◮ e �− → ρ such that Page 4 of 22 A) for each p ∈ V Γ M , α ( E p ) spans Hom( G, Z 2 ) Go Back B) for each e = pq ∈ E Γ M and σ ∈ α ( E p ), the number of times which σ and σ + α ( e ) occur in α ( E p ) is the same as that in α ( E q ). Full Screen Close Quit

Home Page — Abstract definition Let G = ( Z 2 ) k . Title Page We shall work on H ∗ ( BG ; Z 2 ) = Z 2 [ a 1 , ..., a k ] ( ∵ H 1 ( BG ; Z 2 ) ∼ = Contents Hom( G, Z 2 )). Γ n : a connected regular graph of valency n with n ≥ k and no ◭◭ ◮◮ loops. → H 1 ( B ( Z 2 ) k ; Z 2 ) − { 0 } s. t. If there is a map α : E Γ − ◭ ◮ (1) for each vertex p ∈ V Γ , the image α ( E p ) spans H 1 ( B ( Z 2 ) k ; Z 2 ), and Page 5 of 22 (2) for each edge e = pq ∈ E Γ , Go Back � � α ( x ) ≡ α ( y ) mod α ( e ) , x ∈ E p − E e y ∈ E q − E e Full Screen then the pair (Γ , α ) is called a coloring graph of type ( k, n ). Close Quit

Home Page — Examples Title Page a 3 (Γ , α 1 ) is a coloring graph Contents → H 1 ( B ( Z 2 ) 3 ; Z 2 ) α 1 : E Γ − a 1 a 2 a 1 a 2 ◭◭ ◮◮ a 3 where H ∗ ( B ( Z 2 ) 3 ; Z 2 ) = Z 2 [ a 1 , a 2 , a 3 ]. ◭ ◮ Page 6 of 22 a 3 (Γ , α 2 ) is not a coloring graph Go Back a 1 + a 2 a 1 a 2 a 1 ∵ a 1 a 2 �≡ a 1 ( a 1 + a 2 ) mod a 3 Full Screen a 3 Close Quit

Home Page —Faces Title Page (Γ , α ): a coloring graph of type ( k, n ). Γ ℓ : a connected ℓ -valent subgraph of Γ where 0 ≤ ℓ ≤ n . Contents If (Γ ℓ , α | Γ ℓ ) satisfies ◭◭ ◮◮ a) for any two vertices p 1 , p 2 of Γ ℓ , α (( E | Γ ℓ ) p 1 ) and α (( E | Γ ℓ ) p 2 ) ◭ ◮ span the same subspace of H 1 ( BG ; Z 2 ); b) for each edge e = pq ∈ E | Γ ℓ , Page 7 of 22 � � α ( x ) ≡ α ( y ) mod α ( e ) x ∈ ( E | Γ ℓ ) p − ( E | Γ ℓ ) e y ∈ ( E | Γ ℓ ) q − ( E | Γ ℓ ) e Go Back then (Γ ℓ , α | Γ ℓ ) is an ℓ -face of (Γ , α ). Full Screen Close Quit

Home Page Example Title Page Contents a 2 a 1 a 2 a 3 is a 2-face a 1 ◭◭ ◮◮ a 1 + a 2 a 2 + a 3 a 1 + a 2 ◭ ◮ a 1 + a 3 Page 8 of 22 a coloring graph (Γ , α ) Go Back a 1 a 3 is not a 2-face Full Screen a 1 + a 2 a 2 + a 3 Close Quit

Home Page Assumption — Case: valency n of Γ = rank k of G = ( Z 2 ) k Title Page (Γ , α ): a coloring graph of type ( n, n ) with Γ connected. F (Γ ,α ) : the set of all faces of (Γ , α ). Contents — An application for the n -connectedness of a graph. ◭◭ ◮◮ Theorem (Whitney) A graph Γ with at least n + 1 vertices ◭ ◮ is n -connected if and only if every subgraph of Γ , obtained by omitting from Γ any n − 1 or fewer vertices and the edges Page 9 of 22 incident to them, is connected. Theorem (Z. L¨ u and M. Masuda). Suppose that (Γ , α ) is a Go Back coloring graph of type ( n, n ) with Γ connected. If the inter- section of any two faces of dimension ≤ 2 in F (Γ ,α ) is either Full Screen connected or empty, then Γ is n -connected. Close Quit

Home Page Example Title Page a 1 a 2 a 3 a 3 a 2 Contents a 2 + a 3 a 2 + a 3 a 1 + a 3 a 1 + a 2 a 1 + a 2 a 1 + a 3 ◭◭ ◮◮ a 1 ◭ ◮ a 1 a 1 a 2 a 2 a 3 a 3 Page 10 of 22 a 1 + a 2 a 1 + a 2 a 1 + a 3 a 1 + a 3 Go Back a 1 a 1 a 2 + a 3 a 2 + a 3 a 2 a 3 a 1 + a 3 a 1 + a 2 a 3 a 2 Full Screen a 1 + a 2 a 1 + a 3 a 2 + a 3 a 2 + a 3 Close Quit

Home Page § 4. Geometric realization Title Page (Γ , α ):a coloring graph of type ( n, n )= ⇒F (Γ ,α ) = ⇒|F (Γ ,α ) | Contents Example 1. four 2-faces: ◭◭ ◮◮ a 1 a 1 a 1 + a 3 a 1 + a 2 a 1 + a 3 a 1 ◭ ◮ a 1 + a 2 a 1 + a 2 a 1 + a 3 a 2 a 3 a 2 a 3 Page 11 of 22 a 2 + a 3 a 2 + a 3 a 2 a 3 Go Back (Γ , α ):a coloring graph of type (3 , 3) a 2 + a 3 The geometric realization |F (Γ ,α ) | = S 2 Full Screen Close Quit

Home Page Example 2. Title Page a 1 Contents a 3 a 1 a 3 a 2 a 2 a 3 a 3 a 3 ◭◭ ◮◮ a 2 a 2 a 3 a 1 a 1 ◭ ◮ (Γ , α ):a coloring graph of type (3 , 3). Page 12 of 22 a 1 a 2 Go Back The geometric realization |F (Γ ,α ) | = R P 2 a 2 Full Screen a 1 Close Quit

Home Page Generally, Title Page Fact. F (Γ ,α ) forms a simplicial poset of rank n with respect to reversed inclusion with (Γ , α ) as smallest Contents element. ◭◭ ◮◮ ⇓ |F (Γ ,α ) | is a pseudo manifold. ◭ ◮ poset means partially ordered set Page 13 of 22 A poset P is simplicial if it contains a smallest element Go Back ˆ 0 and for each a ∈ P the segment [ˆ 0 , a ] is a boolean algebra (i.e., the face poset of a simplex with empty set Full Screen as the smallest element). Close Quit

Home Page P : a simplicial poset Title Page ⇓ a simplicial cell complex K P in the following way: Contents for each a � = ˆ 0 in P , one obtains a geometrical simplex ◭◭ ◮◮ such that its face poset is [ˆ 0 , a ], and then one glues ◭ ◮ all obtained geometrical simplices together according to the ordered relation in P , so that one can get a cell Page 14 of 22 complex as desired. Go Back By |P| one denotes the underlying space of this cell complex, and one calls |P| the geometric realization Full Screen of P . Close Quit

Home Page Basic problems: Title Page (I). Under what condition, is the geometric realization Contents |F (Γ ,α ) | a closed topological manifold? ◭◭ ◮◮ (II). For any closed topological manifold M n , is there ◭ ◮ a coloring graph (Γ , α ) of type ( n + 1 , n + 1) such that M n ≈ |F (Γ ,α ) | ? Page 15 of 22 Go Back Full Screen Close Quit

Home Page Basic problem (I) Title Page (Γ , α ): a coloring graph of type ( n, n ) with Γ con- nected. Contents The case n = 1: |F (Γ ,α ) | ≈ S 0 ◭◭ ◮◮ The case n = 2: it is easy to see that for any coloring graph (Γ , α ) of type (2 , 2), the geometric realization ◭ ◮ |F (Γ ,α ) | is always a circle. Page 16 of 22 The case n = 3: Go Back Fact. |F (Γ ,α ) | is a closed surface S . Full Screen Close Quit

Home Page Generally, if n > 3, the geometric realization |F (Γ ,α ) | Title Page is not a closed topological manifold. For example, see the following coloring graph (Γ , α ) of type (4 , 4). Contents a 3 a 2 a 1 ◭◭ ◮◮ a 1 a 3 a 2 ◭ ◮ a 4 a 4 a 4 a 4 Page 17 of 22 a 1 a 2 a 3 a 2 a 3 a 1 Go Back χ ( |F (Γ ,α ) | ) = 5 − 12 + 16 − 8 = 1 � = 0 Full Screen so |F (Γ ,α ) | is not a closed topological 3-manifold. Close Quit

Home Page The case n = 4 . Title Page Write v = | V Γ | and e = | E Γ | so 2 v = e . Contents f : the number of all 2-faces in F (Γ ,α ) f 3 : the number of all 3-faces in F (Γ ,α ) ◭◭ ◮◮ Theorem. Let n = 4 . |F (Γ ,α ) | is a closed con- ◭ ◮ nected topological 3-manifold ⇐ ⇒ f = f 3 + v . Page 18 of 22 Problem: for n > 4, to give a sufficient (and neces- Go Back sary) condition that |F (Γ ,α ) | is a closed connected topo- logical manifold. Full Screen Close Quit

Home Page Basic problem (II) Title Page M n : n -dim closed connected topological manifold Contents 1-dim case: M 1 ≈ S 1 . ◭◭ ◮◮ S 1 is realizable by any coloring graph of type (2 , 2). ◭ ◮ Page 19 of 22 2-dim case: Go Back Prop. Any closed surface can be realized by some coloring graph of type (3 , 3) . Full Screen Close Quit

Home Page 3-dim case: Title Page Conjecture: Any closed 3-manifold M 3 is Contents geometrically realizable by a coloring graph (Γ , α ) of type (4 , 4) , i.e., ◭◭ ◮◮ M 3 ≈ |F (Γ ,α ) | . ◭ ◮ 4-dim case: It is well known that there exist closed topological 4-manifolds that don’t admit any trian- Page 20 of 22 gulation. ⇓ Go Back ∃ closed topological 4-manifolds that cannot be real- ized by any coloring graph of type (5 , 5). Full Screen Close Quit

Home Page Proposition. Let M n be a closed manifold. If Title Page M n admits a simplicial cell decomposition with at least n + 2 vertices, then M n can be geometrically Contents realizable by a coloring graph. ◭◭ ◮◮ ◭ ◮ Page 21 of 22 Go Back Full Screen Close Quit