1. MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S14-04-04, matwml@nus.edu.sgmatwml@nus.edu.sg http://math.nus.edu.sg/~matwml Lecture 1. Simplicial Concepts (11, 14 August 2009)

2. Algebra natural, integer (Zahlen), rational (quotient), real, and complex numbers. Hint: use an equivalence relation on denote the Question 1. How are these numbers and their Algebraic operations constructed from

3. What is a Topological Space ? Definition A topological space is a pair where collection of subsets of X (called open subsets) 2. that satisfies the following three properties: 1. (called a topology on X), is a 3. The union the elements in each subset of is in Question 2. Express condition 3 using set theory.

4. What is a Quotient Space ? A topological space,a set we associate a Definition If topology surjection on topology, by called the quotient and a is a topological space andis an equivalence relation on the associated quotient topology is whereis the set of equivalence subsets and is given by Question 3. Describe the quotient topology if http://en.wikipedia.org/wiki/Quotient_space

5. Euler’s Homomorphism and induces, by the first homomorphism theorem for groups, an isomorphism between This isomorphism is also a homeomorphism between the topological space,regarded as a subspace ofwith its usual topology, and the quotient topology oninduced by the canonical homomorphism that is defined by

6. Algebraic Invariants When we speak of a topological space we equivalence class of all topological spaces that are homeomorphic to that space. will often mean (perhaps implicitly) the Algebraic topology studies topological spaces by associating algebraic invariants to spaces. #cc(X) = number of connected components Question 4. Ishomeomorphic to?

7. Affine Space of Dimension n set of points of the affine space ongroup action of (an abelian group under addition) this means that real vector space of dimension n Question 5 Prove The group action is both free and transitive Question 7 Show that every finite dimensional real vector space is an affine space. http://en.wikipedia.org/wiki/Affine_space Question 6 What does free, transitive mean?

8. Affine Combinations for and we observe that the last condition on the the preceding page ensures that Convention: we will write We defineto be that unique Definition For Question 8. Show this point is independent the affine combination with of the ordering of the elements denote the point

9. Affine Maps andare affine spaces, a mapIf is affine if it preserves affine combinations, i.e. Question 9. Prove that if spaces then a map andare vector is affine iff there exists a linear map and such that Question 10. Show that an affine space has a unique topology such that there exists an affine bijection & homeomorphism with

10. Convex Combinations and Simplices An convex combination is an affine combination whose coefficients are nonnegative. Letbe an n-dimensional affine space Points are linearly independent. are in general position (or geometrically independent) if the vectors 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)-dimensional closed ball in

11. Compact Surfaces as Subspaces Some compact surfaces are homeomorphic to subspaces of discannulus2-simplexrectangle others cannot but are homeomorphic to subspaces of (sphere, torus) Real Projective Space to a (topological) subspace of Question 12. What is real projective space? is not homeomorphic

12. Compact Surfaces as Quotient Spaces relate corresponding points on left and right sides torus Question 13. What points are related to obtain a annulus torus? A sphere ? A Klein bottle ? Draw figures.

13. Euler Characteristic of Surfaces Divide the surface of a sphere into polygonal regions having v vertices, e edges, and f faces. Compute the quantity Question 14. Compute v, e, f and for the 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.

14. Barycentric Coordinates If clearly is an n-dimensional affine space and are in general position then and the affine subspace of The k-simplex they span is denoted by they spanned is denoted by are the barycentric coordinates of Question 17. Show they are unique&continuous.

15. Boundary of a Simplex are in general position we define the interior of the simplex Question 18. Show that if as a subspace of the topological space then these two concepts If coincide with the standard topological concepts and boundary is regarded Question 19. Show thatis a disjoint union ofand the interiors of simplices spanned by each subset of

16. Simplicial Maps and is an affine space are in general position then Question 20. Prove that if the set of verticesis determined by the simplex Definition Ifandare affine spaces and are simplices then a map is a simplicial map if there exists an affine mapsuch that

17. Geometric Simplicial Complexes Example Definition Faces of a simplex are the simplices spanned by its proper subsets of vertices. Definition A geometric simplicial complex is a 1. contains each face of each element collection of simplices in an affine space that 2. The intersection of each pair of elements is either empty or a common face http://en.wikipedia.org/wiki/Simplicial_complex

18. Topological Simplicial Complexes is a finite geometric simplicialDefinition If complex in an affine spacewe define its and the associated topological simplicial complex to the be equivalence class of topological spaces that are homeomorphic towith the subspace topology. Assignment: Read pages 1-14 in WuJie and do exercises 2.1, 2.2 on page 25 for finite complexes polyhedron