introduction to algebraic topology -...

Introduction to Algebraic Topology

Jafar Shaffaf

Department of Mathematical Sciences

Shahid Beheshti University, G.C.Tehran, Iran

Workshop On Topological CombinatoricsShahid Beheshti University, G.C.

Tehran, Iran

Wednesday, October 21, 2009

Topological space

I A topological space is a pair (X,O), where X is a (typicallyinfinite) ground set and O is a set system, whose members arecalled the open sets, such that . ∅ ∈ O and X ∈ O, theintersection of finitely many open sets is an open set, and sois the union of an arbitrary collection of open sets.

I A homeomorphism of topological spaces (X1,O1) and(X2,O2) is a bijection ϕ : X1 −→ X2 such that for everyU ⊂ X1, φ(U) ∈ O2 if and only if U ∈ O1 . In other words, abijection is a homeomorphism if and only if both φ, φ−1 arecontinuous.

Topological space

I A topological space is a pair (X,O), where X is a (typicallyinfinite) ground set and O is a set system, whose members arecalled the open sets, such that . ∅ ∈ O and X ∈ O, theintersection of finitely many open sets is an open set, and sois the union of an arbitrary collection of open sets.

I A homeomorphism of topological spaces (X1,O1) and(X2,O2) is a bijection ϕ : X1 −→ X2 such that for everyU ⊂ X1, φ(U) ∈ O2 if and only if U ∈ O1 . In other words, abijection is a homeomorphism if and only if both φ, φ−1 arecontinuous.

Deformation Retract

I If X is a space and Y ⊂ X a subspace of it, a deformationretraction of X onto Y is a family {ft}t∈[0,1] of continuousmaps ft : X −→ X (we can think of t as time), such that

1. f0 is the identity map on X,2. ft(y) = y for all y ∈ Y and all t ∈ [0, 1] (Y remains

stationary), and3. f1(X) = Y

Deformation Retract

1. f0 is the identity map on X,

2. ft(y) = y for all y ∈ Y and all t ∈ [0, 1] (Y remainsstationary), and

3. f1(X) = Y

Deformation Retract

stationary), and

3. f1(X) = Y

Deformation Retract

Homotopy Equivalent

I Two continuous maps f, g : X −→ Y are homotopic (writtenf ∼ g) if there is a continuous interpolation between them;that is, a family {ft}t∈[0,1] of maps ft : X −→ Y dependingcontinuously on t (i.e., the associated bivariate mappingF (x, t) := ft(x) is a continuous map X × [0, 1] −→ Y ,similar to deformation retraction above) such that f0 = f andf1 = g.

Homotopy Equivalent

I Two continuous maps f, g : X −→ Y are homotopic (writtenf ∼ g) if there is a continuous interpolation between them;that is, a family {ft}t∈[0,1] of maps ft : X −→ Y dependingcontinuously on t (i.e., the associated bivariate mappingF (x, t) := ft(x) is a continuous map X × [0, 1] −→ Y ,similar to deformation retraction above) such that f0 = f andf1 = g.

Homotopy Equivalent

I Two spaces X and Y are homotopy equivalent (or have thesame homotopy type) if there are continuous mapsf : X −→ Y and g : Y −→ X such that the compositionf ◦ g : Y −→ Y is homotopic to the identity map idY andg ◦ f ∼ idX .

Simplicial Complexes

I Let v0, v1, . . . , vk be points in Rd. They called affinelydependent if there are real numbers α0, α1, . . . , αk, not all ofthem 0, such that

∑ki=0 αivi = 0 and

∑ki=0 αi = 0.

Otherwise, v0, v1, . . . , vk are called affinely independent.

I The points v0, v1, . . . , vk are affinely independent if and only ifv1 − v0, v2 − v0, . . . , vk − v0 are linearly independent.

I Let v0, v1, . . . , vk be points in Rd. They called affinelydependent if there are real numbers α0, α1, . . . , αk, not all ofthem 0, such that

∑ki=0 αivi = 0 and

∑ki=0 αi = 0.

Otherwise, v0, v1, . . . , vk are called affinely independent.

I The points v0, v1, . . . , vk are affinely independent if and only ifv1 − v0, v2 − v0, . . . , vk − v0 are linearly independent.

I A simplex σ is the convex hull of a finite affinely independentset A in Rd. The points of A are called the vertices of σ. Thedimension of σ is dim(σ) := |A| − 1.

I Examples of Simplices- Here are examples of simplices: apoint, a line segment, a triangle, and a tetrahedron: Theseexamples have dimensions 0, 1, 2, and 3, respectively.

I A simplex σ is the convex hull of a finite affinely independentset A in Rd. The points of A are called the vertices of σ. Thedimension of σ is dim(σ) := |A| − 1.

I Examples of Simplices- Here are examples of simplices: apoint, a line segment, a triangle, and a tetrahedron: Theseexamples have dimensions 0, 1, 2, and 3, respectively.

Geometric Simplicial Complexes

Geometric Simplicial ComplexesI A simplex σ is the convex hull of a finite affinely independent

set A in Rd. The points of A are called the vertices of σ. Thedimension of σ is dim(σ) := |A| − 1.

I A nonempty family ∆ of simplices is a simplicial complex if:

1. Each face of any simplex σ ∈ ∆ is also a simplex of σ .2. The intersection σ1 ∩ σ2 of any two simplices σ1, σ2 ∈ δ

is a face of both σ1 and σ2.

I The union of all simplices in ∆ is the polyhedron of ∆ anddenoted by ‖ ∆ ‖.

1. Each face of any simplex σ ∈ ∆ is also a simplex of σ .

2. The intersection σ1 ∩ σ2 of any two simplices σ1, σ2 ∈ δis a face of both σ1 and σ2.

I Examples of Simplicial Complexes- Zero-dimensionalsimplicial complexes are just configurations of points, while1-dimensional simplicial complexes correspond to graphs(represented geometrically with straight edges that do notcross).

I The following picture shows one 2-dimensional simplicialcomplex in the plane and two cases of putting simplicestogether in ways forbidden by the definition of a simplicialcomplex:

Simplicial Sub-Complexes

I It is a FACT that the set of all faces of a simplex is asimplicial complex.

I A subcomplex of a simplicial complex ∆ is a subset of ∆ thatis itself a simplicial complex (that is, it is closed under takingfaces).

I An important example of a subcomplex is the k-skeleton of asimplicial complex ∆. It consists of all simplices of ∆ ofdimension at most k, and we denote it by ∆≤k

Triangulation

I Let X be a topological space. A simplicial complex ∆ suchthat X = ||∆|| if one exists, is called a triangulation of X

I The simplest triangulation of the sphere Sn−1 is the boundaryof an n-simplex, that is, the subcomplex of σ obtained bydeleting the single n-dimensional simplex (but retaining all ofits proper faces). Indeed, the boundary of an n-simplex ishomeomorphic to Sn−1, as can be seen using the centralprojection:

Triangulation

I Let X be a topological space. A simplicial complex ∆ suchthat X = ||∆|| if one exists, is called a triangulation of X

I The simplest triangulation of the sphere Sn−1 is the boundaryof an n-simplex, that is, the subcomplex of σ obtained bydeleting the single n-dimensional simplex (but retaining all ofits proper faces). Indeed, the boundary of an n-simplex ishomeomorphic to Sn−1, as can be seen using the centralprojection:

Abstract Simplicial Complexes

I Definition. A finite abstract simplicial complex is a finite setA together with a collection ∆ of subsets of A such that ifX ∈ ∆ and Y ⊆ X then Y ∈ ∆.

I The element v ∈ A such that {v} ∈ ∆ is called a vertex of ∆We denote the set of all vertices of ∆ by V (∆) When ∆consists of all subsets of A, it is called a simplex.

I Example The collection of setsK = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}} is an abstractsimplicial complex. To obtain the simplex ∆{1,2,3} one wouldneed to add to this collection the set {1, 2, 3}.

I In this situation we call ∆ a geometric realization of K.

I Each geometric simplicial complex ∆ determines an abstractsimplicial complex. The points of the abstract simplicialcomplex are all vertices of the simplices of ∆, so we setV := V (∆), and the sets in the abstract simplicial complexare just the vertex sets of the simplices of ∆

I For example, for the geometric simplicial complex

we have the abstract simplicial complex{∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}}.

I Each geometric simplicial complex ∆ determines an abstractsimplicial complex. The points of the abstract simplicialcomplex are all vertices of the simplices of ∆, so we setV := V (∆), and the sets in the abstract simplicial complexare just the vertex sets of the simplices of ∆

I For example, for the geometric simplicial complex

we have the abstract simplicial complex{∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}}.

Deformation Retraction for Simplicial Complexes

Deformation Retraction for SimplicialComplexes

I Deformation retraction for topological spaces has a naturaldescription when we are working with simplicial complexes interms of an operation called Collapse

I Definition.Let ∆ be a simplicial complex. Let σ, τ ∈ ∆ suchthat

1. τ ⊂ σ in particular dim τ ≤ dim σ2. σ is a maximal simplex, and no other maximal simplex

contains τ .

I A simplicial collapse of ∆ is the removal of all simplices γsuch that τ ⊆ γ ⊆ σ.

contains τ .

1. τ ⊂ σ in particular dim τ ≤ dim σ

2. σ is a maximal simplex, and no other maximal simplexcontains τ .

contains τ .

I When ∆1 and ∆2 are two simplicial complexes such thatthere exists a sequence of collapses leading from ∆1 to ∆2,this is shown by the notation ∆1 ↘ ∆2.

I Theorem. A sequence of collapses yields a strongdeformation retraction, in particular, a homotopy equivalence.

I When ∆1 and ∆2 are two simplicial complexes such thatthere exists a sequence of collapses leading from ∆1 to ∆2,this is shown by the notation ∆1 ↘ ∆2.

I Theorem. A sequence of collapses yields a strongdeformation retraction, in particular, a homotopy equivalence.

Simplicial Mappings

I Simplicial mappings. It is a fact that a geometric realizationis unique up to homeomorphism. At this occasion we have thevery important notion of a simplicial mapping, which is acombinatorial counterpart of a continuous mapping.

I Definition. Let K and L be two abstract simplicial complexes.A simplicial mapping of K into L is a mappingf : V (K) → V (L) that maps simplices to simplices, i.e., suchthat f(F ) ∈ L whenever F ∈ K.

Simplicial Mappings

I Simplicial mappings. It is a fact that a geometric realizationis unique up to homeomorphism. At this occasion we have thevery important notion of a simplicial mapping, which is acombinatorial counterpart of a continuous mapping.

I Definition. Let K and L be two abstract simplicial complexes.A simplicial mapping of K into L is a mappingf : V (K) → V (L) that maps simplices to simplices, i.e., suchthat f(F ) ∈ L whenever F ∈ K.

Simplicial Mappings

I For every simplicial mapping f : V (∆1) → V (∆2) , it inducesa natural continuous function ||f || : ||∆1|| → ||∆2|| by setting:

||f || :∑

vi∈V (∆1)

tivi 7−→∑

vi∈V (∆1)

tif(vi)

This is easy to check that this map is a continuous mapbetween spaces. If f is injective, then ||f || : ∆1 → ∆2 isinjective too, and if f is an isomorphism, then ||f || is ahomeomorphism.

I Simplicial complexes are connection betweenCombinatorics and Topology.

I We summarize this connection :

Simplicial Mappings

||f || :∑

vi∈V (∆1)

tivi 7−→∑

vi∈V (∆1)

tif(vi)

Simplicial Mappings

||f || :∑

vi∈V (∆1)

tivi 7−→∑

vi∈V (∆1)

tif(vi)

Some Connections

1. Every finite hereditary set system (V,K) can be regarded asan abstract simplicial complex, and it specifies a topologicalspace ||K||(the polyhedron of a geometric realization) up tohomeomorphism.

2. Simplicial maps of simplicial complexes yield continuous mapsof the corresponding spaces.

3. Conversely, if a topological space admits a triangulation, itcan be described purely combinatorially by an abstractsimplicial complex. (This description is not unique)

4. A continuous map, even between triangulated spaces,generally cannot be described by a simplicial map, but it istrue under suitable conditions:

Some Connections

1. there are theorems stating that under suitable conditions, acontinuous map is homotopic to a simplicial map betweensufficiently fine triangulations of the considered spaces, and itcan be approximated by such simplicial maps with anyprescribed precision.

2. These kinds of theorems called simplicial approximationtheorem.

3. In fact there exists a large variety of complexes whosedescription is purely combinatorial. In the following slides wesurvey different situations in which complexes defined bycombinatorial data arise

Some Connections

Flag Complexes

I For a graph G and a subset S ⊂ V (G) of its vertices, we letG[S] denote the corresponding induced graph.

I Definition. Given an arbitrary graph G, we let Cl(G) denotethe abstract simplicial complex whose set of vertices is V (G)and whose simplices are all subsets S ⊂ V (G) such that G[S]is a complete graph. This complex is also called the cliquecomplex of the graph G.

Flag Complexes

I For a graph G and a subset S ⊂ V (G) of its vertices, we letG[S] denote the corresponding induced graph.

I Definition. Given an arbitrary graph G, we let Cl(G) denotethe abstract simplicial complex whose set of vertices is V (G)and whose simplices are all subsets S ⊂ V (G) such that G[S]is a complete graph. This complex is also called the cliquecomplex of the graph G.

Independence Complexes

I Given a graph G, its complement G is the graph with thesame set of vertices such that (v, w) is an edge of G if andonly if v 6= w and (v, w) is not an edge of G. A set of verticesS ⊂ V (G) is called independent if for all v, w ∈ S we have(v, w) is not in E(G).

I Definition. For an arbitrary graph G, the independencecomplex of G, called Ind(G), is the abstract simplicialcomplex whose set of vertices is V (G) and whose simplicesare all the independent sets (anticliques) of G.

I Since independent sets of G are the same as the cliques of G,we see that Ind(G) is isomorphic to Cl(G) as an abstractsimplicial complex.

Order Complex

I By a partially ordered set (poset) we mean a pair (P,�) inwhich P is a set and a binary relation � on P which isreflexive and and transitive.

I Definition. The order complex of a poset P is the simplicialcomplex ∆(P ), whose vertices are the elements of P andwhose simplices are all chains (i.e., linearly ordered subsets, ofthe form {x1, x2, ..., xk}, x1 ≺ x2 ≺ . . . ≺ xk) in P .

I The face poset of a simplicial complex K is the poset P (K),which is the set of all nonempty simplices of K ordered byinclusion.

Order Complex

I For example, the simplicial complex

I has the face poset

Order Complex

I For example, the simplicial complex

I has the face poset

Order Complex

I Let n ∈ N. The partition lattice Πn is the partially ordered setwhose elements are all set partitions of the set [n], and thepartial order is that of refinement. The partition lattice has aminimal element {1}{2} . . . {n} and a maximal element [n].See the figure below. It is a theorem which asserts that∆(Πn) is homotopy equivalent to a wedge of (n− 1)! copiesof Sn−2.

Order Complex

I Let n ∈ N. The partition lattice Πn is the partially ordered setwhose elements are all set partitions of the set [n], and thepartial order is that of refinement. The partition lattice has aminimal element {1}{2} . . . {n} and a maximal element [n].See the figure below. It is a theorem which asserts that∆(Πn) is homotopy equivalent to a wedge of (n− 1)! copiesof Sn−2.

Order Complex

I Example. For a group G and a prime p, we denote by Sp(G)the poset of all nontrivial p-subgroups of G, i.e., thesubgroups whose cardinality is a power of p. Furthermore, welet Ap(G) denote the poset of all nontrivial elementaryabelian p-subgroups of G.

I It turns out that the order simplicial complex ∆(Ap(G)) ishomotopy equivalent to ∆(Sp(G)). This has led to severalinvestigations of these and other posets of various families ofsubgroups of a given group G.

I It is interesting to understand the connections between thegroup-theoretic properties of the considered families ofsubgroups and the topological properties of the ordercomplexes of the corresponding posets.

Order Complex

I Barycentric Subdivision. For a simplicial complex K, thesimplicial complex sd(K) := ∆(P (K)) is called the (first)barycentric subdivision of K.

I More explicitly, the vertices of sd(K) are the nonemptysimplices of K, and the simplices of sd(K) are chains ofsimplices of K ordered by inclusion.

Order Complex

Neighborhood Complex

I Let G be a graph. The neighborhood complex of G is theabstract simplicial complex N (G) defined as follows: itsvertices are all non-isolated vertices of G, and its simplices areall the subsets of V (G) that have a common neighbor.

I Let N(v) denote the set of neighbors of v, i.e.,

N(v) = x ∈ V (G)|(v, x) ∈ E(G).

Then the maximal simplices of N (G) are precisely themaximal elements N(v), for v ∈ V (G).

I Furthermore, for an arbitrary subset A ⊂ V (G), we let N(A)denote the set of common neighbors of A, i.e.N(A) =

⋂v∈A N(v).

N(v) = x ∈ V (G)|(v, x) ∈ E(G).

⋂v∈A N(v).

N(v) = x ∈ V (G)|(v, x) ∈ E(G).

⋂v∈A N(v).

Lov´asz Complex

I The definition of N gives an order-reversing mapN : F(N (G)) → F(N (G)). It can be seen that N3 = N andthat N2(A) ⊇ A, for any A ⊂ V (G).

I Definition. The complex Lo(G) = ∆(N (F(N (G)))) iscalled the Lov´asz complex.

I The Lov´asz complex plays an important role in the proof ofthe so-called Kneser conjecture which is an old problem ingraph theory.

I In fact Lov´asz gives a very nontrivial translation of theexpression ” the chromatic number of the graph G is notequal to k” in the realm of graph theory to the expression”the simplicial complex Lo(G) is k-connected” in the realm of(algebraic) topology.

Lov´asz Complex

Thank You!

introduction to algebraic topology -...

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