108_MA5209 Algebraic Topology

MA5209 Algebraic Topology Lecture Simplicial Concepts (11, 14 August 2009) Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sg http://math.nus.edu.sg/~matwml Algebra N = {1,2,3, } , Z, Q, R, C denote the natural, integer (Zahlen), rational (quotient), real, and complex numbers Question How are these numbers and their Algebraic operations constructed from (N,+ ) ? Hint: use an equivalence relation on N × N Z = {[(a, b)] : (a, b) ∈ N × N} [(a, b)] = {(m, n) ∈ N × N : a + n = b + n} [(a, b)] + [(c, d)] = [(a + c, b + d)] N ∋ k ↔ [(k + 1,1)], − k ↔ [(1, k + 1)],0 ↔ [(1,1)] What is a Topological Space ? Definition A topological space is a pair ( X , Top ) where Top (called a topology on X), is a collection of subsets of X (called open subsets) that satisfies the following three properties: φ ∈ Top , X ∈ Top A ∈ Top and B ∈ Top ⇒ A ∩ B ∈ Top The union the elements in each subset of Top is in Top Question Express condition using set theory What is a Quotient Space ? http://en.wikipedia.org/wiki/Quotient_space A topological space ( X , Top X ) ,a set Y , and a surjection g : X → Y , we associate a topology Top Y on Y , called the quotient −1 topology, by O ∈ Top f ⇔ g (O ) ∈ Top Definition If ( X , Top X ) is a topological space and ~ is an equivalence relation on X , the associated quotient topology is (Y , Top Y ) where Y is the set of equivalence subsets and g : X → Y is given by g ( x) = [ x] = {w ∈ X : x ~ w} Question Describe the quotient topology if X = [1,3], usual top.,1 ~ ~ only equiv Euler’s Homomorphism exp(2π i •) : R → Tc = {c ∈ C :| c |= 1} induces, by the first homomorphism theorem for groups, an isomorphism between T = R / Z = R / kernel(exp (2π i•)) and Tc This isomorphism is also a homeomorphism between the topological space Tc ,regarded as a subspace of C with its usual topology, and the quotient topology on T = R / Z induced by the canonical homomorphism π : R → T = R / Z that is defined by π ( x ) = x + Z , x ∈ R Algebraic Invariants When we speak of a topological space we will often mean (perhaps implicitly) the equivalence class of all topological spaces that are homeomorphic to that space Algebraic topology studies topological spaces by associating algebraic invariants to spaces #cc(X) = number of connected components Question Is homeomorphic to ? Affine Space of Dimension n http://en.wikipedia.org/wiki/Affine_space A set of points of the affine space V real vector space of dimension n (an abelian group under addition) τ : A × V → A group action of V on A this means that τ ( p,0) = p, ∀p ∈ A τ (τ ( p, u ), v) = τ ( p, u + v), ∀p ∈ A, u , v ∈ V The group action is both free and transitive Question Prove τ (τ ( p, u ),−u ) = p, ∀p ∈ A, u ∈ V Question What does free, transitive mean? Question Show that every finite dimensional real vector space is an affine space Affine Combinations Convention: we will write p + v for τ ( p, v ) and we observe that the last condition on the the preceding page ensures that ∀p, q ∈ A, ∃!v ∈ V ∋ p + v = q We define q − p to be that unique v ∈ V Definition For p1 , , pk ∈ A, r1 , , rk ∈ R with r1 +  + rk = the affine combination r 1p1 + + rk pk denote the point p1 + r2 ( p2 − p1 ) +  + rk ( pk − p1 ) ∈ A Question Show this point is independent of the ordering of the elements p1 , , pk If A and B Affine Maps are affine spaces, a map f :A→ B is affine if it preserves affine combinations, i.e f (r1 p1 +  + rk pk ) = r1 f ( p1 ) +  + rk f ( pk ) Question Prove that if A and B are vector spaces then a map f : A → B is affine iff there exists a linear map L : A → B and b ∈ B such that f (a ) = La + b, ∀a ∈ A Question 10 Show that an affine space A has a unique topology such that there exists an affine bijection & homeomorphism with R n Convex Combinations and Simplices An convex combination is an affine combination whose coefficients are nonnegative Let A be an n-dimensional affine space Points p1 , , pk ∈ A are in general position (or geometrically independent) if the vectors p2 − p1 , , pk − p1 are linearly independent Then the set of convex combinations is called the (k-1)-simplex spanned by these points Question 11 Show that all (k-1)-simplices are affinely isomorphic and homeomorphic to a k −1 (k-1)-dimensional closed ball in R Compact Surfaces as Subspaces Some compact surfaces are homeomorphic to subspaces of R disc ≠ ≈ ≈ rectangle 2-simplex annulus others cannot but are homeomorphic to subspaces of R (sphere, torus) Real Projective Space RP is not homeomorphic to a (topological) subspace of R Question 12 What is real projective space? Compact Surfaces as Quotient Spaces annulus ≈ a a relate corresponding points on left and right sides b torus a ≈ a b Question 13 What points are related to obtain a torus? A sphere ? A Klein bottle ? Draw figures Euler Characteristic of Surfaces Divide the surface of a sphere into polygonal regions having v vertices, e edges, and f faces χ = v−e+ f Question 14 Compute v, e, f and χ for the Compute the quantity surfaces of each of the the five platonic solids and discuss the results Question 15 Repeat using various triangular divisions of the sphere Question 16 Repeat for other surfaces Hint: use their quotient space representations Barycentric Coordinates If A is an n-dimensional affine space and a0 , , ak ∈ A are in general position then clearly k ≤ n and the affine subspace of they spanned is denoted by A Aff (a0 , , ak ) = {t0 a0 +  + t k ak : ti ∈ R, ∑i =0 ti = 1} k The k-simplex they span is denoted by a0  ak = {t0 a0 +  + t k ak ∈ Aff (a0 ,  , ak ) : ≤ ti } t0 , , t k are the barycentric coordinates of t a0 +  + t k a k ∈ a0  a k Question 17 Show they are unique&continuous Boundary of a Simplex If a0 , , ak ∈ A are in general position we define the interior of the simplex a0  ak Int (a0  ak ) = {t0 a0 +  + t k ak ∈ a0  ak : ∀ti > 0} and boundary ∂a0  ak = a0  ak \ Int ( a0  ak ) Question 18 Show that if a0  a k is regarded as a subspace of the topological space Aff (a0 ,  , ak ), then these two concepts coincide with the standard topological concepts Question 19 Show that a0  ak is a disjoint union of {a0 ,  , ak } and the interiors of simplices spanned by each subset of {a0 ,  , ak } Simplicial Maps Question 20 Prove that if A is an affine space and a0 , , ak ∈ A are in general position then the set of vertices {a0 , , ak }is determined by the simplex a0  ak Definition If A and B are affine spaces and σ ⊆ A,τ ⊆ B are simplices then a map f : σ → τ is a simplicial map if there exists an affine map ~ f :A→ B such that ~ f (vertices(σ )) ⊆ vertices(τ ), ~ f |σ = f Geometric Simplicial Complexes Definition Faces of a simplex are the simplices spanned by its proper subsets of vertices a0 • faces(a0 a1a2 ) = a2 Example a1 • • {a0 a1 , a0 a2 , a1a2 , a0 , a1 , a2 } http://en.wikipedia.org/wiki/Simplicial_complex Definition A geometric simplicial complex is a collection of simplices in an affine space that contains each face of each element The intersection of each pair of elements is either empty or a common face Topological Simplicial Complexes Definition If K is a finite geometric simplicial complex in an affine space A we define its polyhedron |K |= σ ⊂ A σ ∈K and the associated topological simplicial complex to the be equivalence class of topological spaces that are homeomorphic to | K | with the subspace topology Assignment: Read pages 1-14 in WuJie and exercises 2.1, 2.2 on page 25 for finite complexes ... subspace of C with its usual topology, and the quotient topology on T = R / Z induced by the canonical homomorphism π : R → T = R / Z that is defined by π ( x ) = x + Z , x ∈ R Algebraic Invariants... class of all topological spaces that are homeomorphic to that space Algebraic topology studies topological spaces by associating algebraic invariants to spaces #cc(X) = number of connected components... space ( X , Top X ) ,a set Y , and a surjection g : X → Y , we associate a topology Top Y on Y , called the quotient −1 topology, by O ∈ Top f ⇔ g (O ) ∈ Top Definition If ( X , Top X ) is a topological