Representing PL manifolds by edge-colored graphs: basic concepts and - - PowerPoint PPT Presentation
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs: basic concepts and recent results Paola Cristofori University of Modena and Reggio Emilia (Italy)
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds
Paola Cristofori University of Modena and Reggio Emilia (Italy) Conference “Random Geometry and Physics” Paris, October 17-21, 2016
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds
Objects PL n-manifolds = triangulated (topological) manifolds s.t. each vertex has a neighbourhood whose boundary is PL-isomorphic either to the boundary of an n-simplex or to an (n − 1)-simplex Morphisms Piecewise Linear (PL) maps = induced by simplicial maps
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
An n-pseudocomplex is a finite collection of n-simplices (together with their faces) such that two n-simplices may have a union of faces in common. A (pseudo)triangulation of a PL n-manifold M is an n-pseudocomplex K whose points form a topological space |K|, PL-isomorphic to M
Torus Möbius strip
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
K pseudocomplex triangulating a closed PL n-manifold Mn ξ : S0(K) → ∆n = {0, 1, . . . , n} (vertex-labelling) injective on each n-simplex of K (K, ξ) = is a colored triangulation of Mn Remark: If Mn = |K|, then (K ′, ξ) is a colored triangulation, where K ′ first barycentric subdivision of K ξ(v) = r iff v barycenter of an r-simplex of K
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Γ = (V (Γ), E(Γ)) 1-skeleton of the dual cellular complex of K (regular graph of degree n + 1); γ : E(Γ) → ∆n (edge-coloration) defined by: γ(e) = c if e ∈ E(Γ) is dual to an (n − 1)-simplex of K having no c-labelled vertex.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
1 1 1 2
A colored triangulation of the torus
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
1 1 1 2
A B D E F C take a vertex v(σ) for each n-simplex σ of K
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group 1 1 1 2
K Γ
A B D E F C A B F E D C
join two vertices v(σ) and v(σ′) with a c-colored edge (c ∈ ∆n) iff σ and σ′ have in common the (n − 1)-dimensional face opposite to their c-colored vertices.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group 1 1 1 2
K Γ
A B D E F C A B F E D C
join two vertices v(σ) and v(σ′) with a c-colored edge (c ∈ ∆n) iff σ and σ′ have in common the (n − 1)-dimensional face opposite to their c-colored vertices.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group 1 1 1 2
K Γ
A B D E F C A B F E D C
join two vertices v(σ) and v(σ′) with a c-colored edge (c ∈ ∆n) iff σ and σ′ have in common the (n − 1)-dimensional face opposite to their c-colored vertices.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group 1 1 1 2
K Γ
A B D E F C A B F E D C
Γ(K) is an (n+1)-colored graph i.e. adjacent edges have different colors
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
1 1 1 2
K Γ
1 1 2 A B D E F C A B F E D C D A B C D A B C
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Let (Γ, γ) be an (n + 1)-colored graph, 1) take an n-simplex σ(x) for every vertex x ∈ V (Γ), and label its vertices by ∆n; 2) if x, y ∈ V (Γ) are joined by a c-colored edge, identify the (n − 1)-faces of σ(x) and σ(y) opposite to c-labelled vertices, so that equally labelled vertices coincide.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
⋆ K(Γ) is an n-pseudomanifold ⋆ (Γ, γ) represents Mn = |K(Γ)| ⋆ If Mn is a closed manifold, (Γ, γ) is called a gem = “graph encoded manifold” of Mn. Remark: Γ(K(Γ)) = Γ for any Γ, but K(Γ(K)) = K iff the disjoint star
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
CONSEQUENCES: Mn = |K(Γ)| is
iff Γ is bipartite;
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
CONSEQUENCES: Mn = |K(Γ)| is
iff Γ is bipartite; ∀B ⊂ ∆n, with #B = h, there is a bijection between (n − h)-simplices of K(Γ) whose vertices are labelled by ∆n − {B} and connected components of h-colored graph ΓB = (V (Γ), γ−1(B)) (B-residues of Γ). In particular: c-labelled vertices of K(Γ) are in bijection with connected components of Γˆ
c = Γ∆n−{c}.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
CONSEQUENCES: Mn = |K(Γ)| is
iff Γ is bipartite; ∀B ⊂ ∆n, with #B = h, there is a bijection between (n − h)-simplices of K(Γ) whose vertices are labelled by ∆n − {B} and connected components of h-colored graph ΓB = (V (Γ), γ−1(B)) (B-residues of Γ). In particular: c-labelled vertices of K(Γ) are in bijection with connected components of Γˆ
c = Γ∆n−{c}.
|K(Γ)| is a (closed) n-manifold if and only if, for every c ∈ ∆n, each connected component of Γˆ
c represents Sn−1.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
An (n + 1)-colored graph (Γ, γ) is called contracted if ∀c ∈ ∆n, either Γˆ
c
is connected or no connected components of Γˆ
c represents the
(n − 1)-sphere. A crystallization of a closed n-manifold Mn is a contracted gem (Γ, γ) of
vertices ( = minimum possible number of vertices).
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
An (n + 1)-colored graph (Γ, γ) is called contracted if ∀c ∈ ∆n, either Γˆ
c
is connected or no connected components of Γˆ
c represents the
(n − 1)-sphere. A crystallization of a closed n-manifold Mn is a contracted gem (Γ, γ) of
vertices ( = minimum possible number of vertices). Pezzana Existence Theorem (1974) Each PL n-manifold admits a crystallization. Generalization Each polyhedron triangulated by an n-pseudomanifold can be represented by a contracted (n + 1)-colored graph.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Contracted triangulation and dual crystallization of S3
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Crystallizations of S1 × S2 and CP2
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
a c C B A b d D
A contracted 4-colored graph representing S1 × D2
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
h-dipole (1 ≤ h ≤ n) of Γ Θ = (V (Θ) = {v, w}, E(Θ) = {ej1, . . . , ejh}) j1, . . . , jh ∈ ∆n, such that v and w belong to different connected components of Γ∆n−{j1,...,jh}. Γ cancellation − − − − − − − − → Γ′ Γ insertion ← − − − − − Γ′ An h-dipole Θ is called proper if and only if |K(Γ)| and |K(Γ′)| are PL-isomorphic.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Gagliardi, 1987 Let Θ be an h-dipole of an (n + 1)-colored graph (Γ, γ). If at least one of the connected components of Γ∆n−{j1,...,jh} containing v or w represents an (n − h)-sphere then Θ is proper. As a consequence, any dipole of a gem of a closed PL manifold is proper.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Gagliardi, 1987 Let Θ be an h-dipole of an (n + 1)-colored graph (Γ, γ). If at least one of the connected components of Γ∆n−{j1,...,jh} containing v or w represents an (n − h)-sphere then Θ is proper. As a consequence, any dipole of a gem of a closed PL manifold is proper. Any two gems of a closed PL manifold can be obtained from each other by a finite sequence of dipole moves.
In the case of pseudomanifolds this is not generally true.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
First method If (Γ, γ) is a contracted (n + 1)-colored graph. For each pair i, j ∈ ∆n, let πij(Γ) be the group presented in the following way: (*) the generators are all (∆n − {i, j})-residues, but one arbitrarily chosen; (**) the relators are “read” on all {i, j}-colored cycles of Γ, but one arbitrarily chosen (unless n = 2).
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
First method If (Γ, γ) is a contracted (n + 1)-colored graph. For each pair i, j ∈ ∆n, let πij(Γ) be the group presented in the following way: (*) the generators are all (∆n − {i, j})-residues, but one arbitrarily chosen; (**) the relators are “read” on all {i, j}-colored cycles of Γ, but one arbitrarily chosen (unless n = 2). Second method If Γˆ
c is connected, for some color c ∈ ∆n, let πc(Γ) be the group
presented in the following way: (*) the generators are the c-colored edges; (**) the relators are “read” on the {c, k}-colored cycles, for each color k = c.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds Representing PL manifolds by edge-colored graphs Moves Computation of the fundamental group
Proposition Let (Γ, γ) be an (n + 1)-colored graph. Given i, j ∈ ∆n (resp. c ∈ ∆n), then πij(Γ) ∼ = π1(|K(Γ) \ Σˆ
iˆ j|)
(resp. πc(Γ) ∼ = π1(|K(Γ) \ Σˆ
c|)).
In particular, if (Γ, γ) represents a closed n-manifold, then πij(K(Γ) ∼ = π1(|K(Γ)|) (resp. πc(Γ) ∼ = π1(|K(Γ)|)) , for each i, j ∈ ∆n (resp. for each c ∈ ∆n). if (Γ, γ) represents an n-manifold with boundary, then πij(K(Γ) ∼ = π1(|K(Γ)|) (resp. πc(Γ) ∼ = π1(|K(Γ)|)) , iff i, j ∈ ∆n are non-singular colors (resp. c ∈ ∆n is a non-singular color).
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
A cellular embedding i : |Γ| → F of an (n + 1)-colored graph (Γ, γ) into a (closed) surface F is called a regular embedding if there exists a cyclic permutation ε = (ε1, . . . , εn) of ∆n s.t. each connected component of F − i(|Γ|) is an open ball bounded by the image of an {εi, εi+1}− colored cycle of Γ.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
A cellular embedding i : |Γ| → F of an (n + 1)-colored graph (Γ, γ) into a (closed) surface F is called a regular embedding if there exists a cyclic permutation ε = (ε1, . . . , εn) of ∆n s.t. each connected component of F − i(|Γ|) is an open ball bounded by the image of an {εi, εi+1}− colored cycle of Γ.
EXAMPLE: Regular embedding corresponding to ε = (green, red, blue, grey)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Gagliardi, 1981 For each (n + 1)-colored graph (Γ, γ) and for every cyclic permutation ε = (ε1, . . . , εn) of ∆n, there exists a regular embedding of Γ onto a suitable surface Fε. Moreover:
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Gagliardi, 1981 For each (n + 1)-colored graph (Γ, γ) and for every cyclic permutation ε = (ε1, . . . , εn) of ∆n, there exists a regular embedding of Γ onto a suitable surface Fε. Moreover:
Definition The regular genus ρǫ(Γ) of Γ with respect to ε is the classical genus (resp. half of the genus) of the orientable (resp. non-orientable) surface Fε :
gεiεi+1 + (1 − n)p = 2 − 2ρε(Γ)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Definition The regular genus G(Mn) of a closed PL n-manifold Mn is the minimum value of ρε(Γ), for every gem (Γ, γ) of Mn and for every cyclic permutation ε of ∆n: G(Mn) = min
(Γ.γ) gem of Mn ε cyclic permutation of ∆n
Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Definition The regular genus G(Mn) of a closed PL n-manifold Mn is the minimum value of ρε(Γ), for every gem (Γ, γ) of Mn and for every cyclic permutation ε of ∆n: G(Mn) = min
(Γ.γ) gem of Mn ε cyclic permutation of ∆n
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The regular genus is a PL-manifold invariant which extends to arbitrary dimension the classical genus of a surface and the Heegaard genus of a 3-manifold.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The regular genus is a PL-manifold invariant which extends to arbitrary dimension the classical genus of a surface and the Heegaard genus of a 3-manifold. For every PL n-manifold Mn (n ≥ 3), G(Mn) ≥ rk(π1(Mn))
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The regular genus is a PL-manifold invariant which extends to arbitrary dimension the classical genus of a surface and the Heegaard genus of a 3-manifold. For every PL n-manifold Mn (n ≥ 3), G(Mn) ≥ rk(π1(Mn)) For every n ≥ 2, G(Mn) = 0 ⇐ ⇒ Mn ∼ =PL Sn
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The regular genus is a PL-manifold invariant which extends to arbitrary dimension the classical genus of a surface and the Heegaard genus of a 3-manifold. For every PL n-manifold Mn (n ≥ 3), G(Mn) ≥ rk(π1(Mn)) For every n ≥ 2, G(Mn) = 0 ⇐ ⇒ Mn ∼ =PL Sn Remark If (Γ, γ) is an (n + 1)-colored graph s.t. ρε(Γ) = 0, for a cyclic permutation ε of ∆n, then |K(Γ)| is an n-manifold and, by the above result, it is PL-isomorphic to Sn
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Conjecture (Ferri - Gagliardi) Let Mn
1 , Mn 2 be two closed (orientable) PL n-manifolds.
Then, G(Mn
1 #Mn 2 ) = G(Mn 1 ) + G(Mn 2 )
For n = 4, the Conjecture implies the 4-dimensional Smooth Poincar´ e Conjecture, via Wall Theorem on homotopic 4-manifolds:
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Conjecture (Ferri - Gagliardi) Let Mn
1 , Mn 2 be two closed (orientable) PL n-manifolds.
Then, G(Mn
1 #Mn 2 ) = G(Mn 1 ) + G(Mn 2 )
For n = 4, the Conjecture implies the 4-dimensional Smooth Poincar´ e Conjecture, via Wall Theorem on homotopic 4-manifolds: Question Is the regular genus finite-to-one ?
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The Gurau degree of an (n + 1)- colored graph (Γ, γ) is ωG(Γ) =
n! 2
ρε(i)(Γ) where the ε(i)’s are the cyclic permutations of ∆n up to inverse. The Gurau degree DG(Mn) of a closed PL n-manifold Mn is the minimum value of ωG(Γ), for every gem (Γ, γ) of Mn
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
The Gurau degree of an (n + 1)- colored graph (Γ, γ) is ωG(Γ) =
n! 2
ρε(i)(Γ) where the ε(i)’s are the cyclic permutations of ∆n up to inverse. The Gurau degree DG(Mn) of a closed PL n-manifold Mn is the minimum value of ωG(Γ), for every gem (Γ, γ) of Mn DG(Mn) ≥ n! 2 G(Mn)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Definition: Gem-complexity of a closed n-manifold Mn k(Mn) = p − 1, 2p = minimum order of a crystallization of Mn Gem-complexity is always realized by a crystallization with no dipoles. Catalogues of crystallizations for increasing gem-complexity n = 3 orientable case Lins - Casali - Cristofori (from 1995 to 2013): up to gem-complexity 15 (i.e. 32 vertices) n = 3 nonorientable case Bandieri - Casali - Cristofori - Gagliardi (from 1999 to 2009): up to gem-complexity 14 (i.e. 30 vertices) n = 4 Casali - Cristofori (2015): up to gem-complexity 9 (i.e. 20 vertices) Orientable 3-manifolds with toric boundary: Cristofori - Mulazzani - Fominykh - Tarkaev (2016) : contracted bipartite 4-colored graphs with no dipoles up to gem-complexity 5 (i.e. 12 vertices)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Casali - Cristofori 2013 Let M be a closed orientable prime 3-manifold. If c(M) ≤ 6 and M = L(p, q), then k(M) < 5 + 2c(M); if M = L(p, q) and 1 ≤ c(M) ≤ 5, then k(M) = 5 + 2c(M). Remark: k(M) ≤ 5 + 2c(M) holds for closed non-orientable prime 3-manifolds with c(M) ≤ 7, too. Conjecture k(M) ≤ 5 + 2 · c(M) for each closed 3-manifold M. Remark: There exist 3-manifolds for which strict inequality holds.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds The regular genus gem-complexity
Casali - Cristofori - Dartois - Grasselli (preprint) For any (n + 1)-colored graph (Γ, γ) of order 2p: ωG(Γ) = p − 1 −
(gˆ
i − 1) + χ(K(Γ)),
if n = 3 ωG(Γ) = 6(p − 1 −
(gˆ
i − 1) + χ(K(Γ)) − 2),
if n = 4 For any closed PL n-manifold M: DG(M) = k(M), if n = 3 DG(M) = 6(k(M) + χ(M) − 2), if n = 4
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
TOP category topological manifolds, up to homeomorphisms
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
TOP category topological manifolds, up to homeomorphisms PL category PL-manifolds, up to PL-isomorphisms
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
TOP category topological manifolds, up to homeomorphisms PL category PL-manifolds, up to PL-isomorphisms DIFF category smooth manifolds, up to diffeomorphisms
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
n=3 TOP=PL (any topological 3-manifold admits a PL-structure which is unique up to PL-isomorphisms) PL=DIFF (each PL-structure on a 3-manifold is smoothable in a unique way up to diffeomorphisms)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
n=3 TOP=PL (any topological 3-manifold admits a PL-structure which is unique up to PL-isomorphisms) PL=DIFF (each PL-structure on a 3-manifold is smoothable in a unique way up to diffeomorphisms) n=4 PL=DIFF (each PL-structure on a 4-manifold is smoothable in a unique way up to diffeomorphisms) TOP=DIFF
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
n=3 TOP=PL (any topological 3-manifold admits a PL-structure which is unique up to PL-isomorphisms) PL=DIFF (each PL-structure on a 3-manifold is smoothable in a unique way up to diffeomorphisms) n=4 PL=DIFF (each PL-structure on a 4-manifold is smoothable in a unique way up to diffeomorphisms) TOP=DIFF
there are topological 4-manifolds admitting no smooth structure;
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
n=3 TOP=PL (any topological 3-manifold admits a PL-structure which is unique up to PL-isomorphisms) PL=DIFF (each PL-structure on a 3-manifold is smoothable in a unique way up to diffeomorphisms) n=4 PL=DIFF (each PL-structure on a 4-manifold is smoothable in a unique way up to diffeomorphisms) TOP=DIFF
there are topological 4-manifolds admitting no smooth structure; there can be non-diffeomorphic smooth structures on the same topological 4-manifold.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Closed simply-connected oriented topological 4-manifolds are classified by their intersection forms (Freedman, 1982).
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Closed simply-connected oriented topological 4-manifolds are classified by their intersection forms (Freedman, 1982). Intersection forms of closed simply-connected smooth 4-manifolds are only of restricted types (Donaldson, 1983 - Furuta, 2001).
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Closed simply-connected oriented topological 4-manifolds are classified by their intersection forms (Freedman, 1982). Intersection forms of closed simply-connected smooth 4-manifolds are only of restricted types (Donaldson, 1983 - Furuta, 2001). Up to now there is no classification of smooth structures on any given smoothable topological 4-manifold.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Some recent results about different smooth structures on the same TOP manifold: [Akhmedov-Doug Park, 2010], [Akhmedov-Ishida-Doug Park, 2013] There exist (infinitely many) non-diffeomorphic smooth structures on: #2h−1CP2#2hCP2, for any integer h ≥ 1 #2h−1(S2 × S2), for h ≥ 138 #2h−1(CP2#CP2), for h ≥ 23 #2p(S2 × S2) and #2p(CP2#CP2), for large enough integers p not divisible by 4.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Some recent results about different smooth structures on the same TOP manifold: [Akhmedov-Doug Park, 2010], [Akhmedov-Ishida-Doug Park, 2013] There exist (infinitely many) non-diffeomorphic smooth structures on: #2h−1CP2#2hCP2, for any integer h ≥ 1 #2h−1(S2 × S2), for h ≥ 138 #2h−1(CP2#CP2), for h ≥ 23 #2p(S2 × S2) and #2p(CP2#CP2), for large enough integers p not divisible by 4. The existence of exotic PL-structures on S4, CP2, S2 × S2 or CP2#CP2
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Basak - Casali, 2015 k(M4) ≥ 3χ(M4) + 10rk(π1(M4)) − 6 G(M4) ≥ 2χ(M4) + 5rk(π1(M4)) − 4 χ(M4) = Euler characteristic of M4 (closed PL 4-manifold)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Basak - Casali, 2015 k(M4) ≥ 3χ(M4) + 10rk(π1(M4)) − 6 G(M4) ≥ 2χ(M4) + 5rk(π1(M4)) − 4 χ(M4) = Euler characteristic of M4 (closed PL 4-manifold) In the simply-connected case: k(M4) ≥ 3β2(M4) G(M4) ≥ 2β2(M4) where β2(M4) = second Betti number of M4
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Casali - Cristofori, 2015 Let M4 be a simply-connected closed PL 4-manifold. If either k(M4) ≤ 65 or G(M4) ≤ 43, then M4 is TOP-homeomorphic to (#rCP2)#(#r ′CP
2)
#s(S2 × S2), where r + r ′ = β2(M4), s = 1
2β2(M4)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
[Gagliardi, 1989] [Cavicchioli, 1989 - 1992] [Cavicchioli-Meschiari 1993] Let M4 be a closed PL 4-manifold, then, if M4 is orientable: G(M4) = ρ ≤ 3 = ⇒ M4 ∼ =
#ρ−2(S3×S1)#CP2 If M4 is non-orientable: G(M4) = ρ ≤ 2 = ⇒ M4 ∼ = #ρ(S3 ×S1)
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
[Casali, 1996] [Casali-Malagoli, 1997] Let M4 be a closed PL 4-manifold (orientable or not). G(M4) = rk(π1(M4)) = ρ ⇐ ⇒ M4 ∼ = #ρ(S3 ⊗ S1) G(M4) = rk(π1(M4)) = ⇒ G(M4) − rk(π1(M4)) ≥ 2 G(M4) − rk(π1(M4)) = 2 and π1(M4) = ∗mZ ⇐ ⇒ M4 ∼ = #m(S3 ⊗ S1)#CP2 No M4 exists with G(M4) − rk(π1(M4)) = 3 and π1(M4) = ∗mZ.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Every closed PL 4-manifold M4 admits a handle-decomposition M4 = H(0)∪(H(1)
1 ∪· · ·∪H(1) r1 )∪(H(2) 1 ∪· · ·∪H(2) r2 )∪(H(3) 1 ∪· · ·∪H(3) r3 )∪H(4)
p-handle: H(p)
i
= Dp × D4−p (1 ≤ p ≤ 4) attaching map: f (p)
i
: ∂Dp × D4−p → ∂(H(0) ∪ . . . (H(p−1)
1
∪ · · · ∪ H(p−1)
rp−1 ))
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Every closed PL 4-manifold M4 admits a handle-decomposition M4 = H(0)∪(H(1)
1 ∪· · ·∪H(1) r1 )∪(H(2) 1 ∪· · ·∪H(2) r2 )∪(H(3) 1 ∪· · ·∪H(3) r3 )∪H(4)
p-handle: H(p)
i
= Dp × D4−p (1 ≤ p ≤ 4) attaching map: f (p)
i
: ∂Dp × D4−p → ∂(H(0) ∪ . . . (H(p−1)
1
∪ · · · ∪ H(p−1)
rp−1 ))
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Every closed PL 4-manifold M4 admits a handle-decomposition M4 = H(0)∪(H(1)
1 ∪· · ·∪H(1) r1 )∪(H(2) 1 ∪· · ·∪H(2) r2 )∪(H(3) 1 ∪· · ·∪H(3) r3 )∪H(4)
p-handle: H(p)
i
= Dp × D4−p (1 ≤ p ≤ 4) attaching map: f (p)
i
: ∂Dp × D4−p → ∂(H(0) ∪ . . . (H(p−1)
1
∪ · · · ∪ H(p−1)
rp−1 ))
There is a unique way to attach the 3- and 4-handles to the union of 0, 1, 2-handles.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
If (Γ, γ) is a crystallization of a closed M4 and {{r, s, t}, {i, j}} is a partition of the five vertices of the associated pseudocomplex K(Γ), then M4 admits a decomposition of type
M4 = N(r, s, t) ∪φ N(i, j)
where: N(r, s, t) denotes a regular neighborhood of the subcomplex of K(Γ) generated by vertices labelled by {r, s, t} (union of 0,1,2-handles) N(i, j) denotes a regular neighborhood of the subcomplex of K(Γ) generated by vertices labelled by {i, j} (union of 3,4-handles) φ is a boundary identification.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Casali - Cristofori, 2014 Let M4 be a closed PL 4-manifold. Then: k(M4) = 0 ⇐ ⇒ M4 ∼ = S4 k(M4) = 3 ⇐ ⇒ M4 ∼ = CP2 k(M4) = 4 ⇐ ⇒ M4 ∼ = S1 × S3
M4 ∼ = S1 × S3 k(M4) = 6 ⇐ ⇒ M4 ∼ = S2 × S2 or M4 ∼ = CP2#CP2
M4 ∼ = CP2#CP
2
k(M4) = 7 ⇐ ⇒ M4 ∼ = RP4
M4 ∼ = CP2#(S1 × S3)
M4 ∼ = CP2#(S1 × S3) k(M4) = 8 ⇐ ⇒ M4 ∼ = #2(S1 × S3)
M4 ∼ = #2(S1 × S3) no closed PL 4-manifold M4 exists with k(M4) ∈ {1, 2, 5}
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Casali - Cristofori, 2014 Let M4 be a closed PL 4-manifold. Then: k(M4) = 0 ⇐ ⇒ M4 ∼ = S4 k(M4) = 3 ⇐ ⇒ M4 ∼ = CP2 k(M4) = 4 ⇐ ⇒ M4 ∼ = S1 × S3
M4 ∼ = S1 × S3 k(M4) = 6 ⇐ ⇒ M4 ∼ = S2 × S2 or M4 ∼ = CP2#CP2
M4 ∼ = CP2#CP
2
k(M4) = 7 ⇐ ⇒ M4 ∼ = RP4
M4 ∼ = CP2#(S1 × S3)
M4 ∼ = CP2#(S1 × S3) k(M4) = 8 ⇐ ⇒ M4 ∼ = #2(S1 × S3)
M4 ∼ = #2(S1 × S3) no closed PL 4-manifold M4 exists with k(M4) ∈ {1, 2, 5} There are no exotic structures on M4 with k(M4) ≤ 8.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Casali - Cristofori - Gagliardi, 2015 Let M4 ∼ =PL (#rCP2)#(#r ′CP
2)#(#s(S2 × S2))#(#tK3) with r, r ′,
s, t ≥ 0. Then, G(M4) = 2(r + r ′ + 2s + 22t) and k(M4) = 3(r + r ′ + 2s + 22t). In particular: G(K3) = 44 and k(K3)=66.
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Kronheimer - Mrowka, 1994 K3#CP2 ≇PL 3CP2#20CP2 There exist (minimal) crystallizations of K3
Crystallizations of homeomorphic, but not PL-isomorphic closed 4-manifolds have been obtained
Paola Cristofori Representing PL manifolds by edge-colored graphs
Basic crystallization theory PL invariants via colored graph Crystallizations of 4-manifolds TOP classification in the simply-connected case PL classification
Paola Cristofori Representing PL manifolds by edge-colored graphs