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 2. Homotopy Concepts (18, 21 August 2009)

2. Homotopy Definition Let Remark All maps between topological spaces will be assumed to be continuous unless explicitly stated that they are not! be topological spaces and have the product topology. A map is homotopic to a map If there exists a map(a homotopy) such that: http://en.wikipedia.org/wiki/Homotopy Question 1. Show this is an equivalence relation on the set of continuous functions from X to Y. Let [X,Y] denote the set of equivalence classes.

3. Example 1. Proposition 1. Let X be a topological space and Y a subspace of an affine space and f, g : X  Y be maps such that for every x in X, the line segment connecting f(x) and g(x) lies in Y. Then f is homotopic to g. Proof Construct the homotopy Corollary 1 Let X be a topological space and let by be maps with Then f is homotopic to g. Question 2. Prove this corollary.

4. Example 2. Letand For be a map. construct maps Theorem Proof Construct is homotopic to by Then

5. Example 2. Letand be a map defined by a polynomial with complex Then for every constant function by is homotopic to theProposition 2. Each coefficients and such that define the map Proof Defineby Proposition 3. For sufficiently large is homotopic to the map Question 3. Prove this using Corollary 1.

6. Retractions Definition If Y is a topological subspace of X, a retraction of X onto Y is a map r : X  Y such that the restriction Example X = annulus Y = circle http://en.wikipedia.org/wiki/Deformation_retract

7. Retractions as shown in the illustration below illustration for n = 2 Theorem A map to itself that has no fixed points induces a retraction from the n+1 ball Question 4. Prove that there exists such a map r so

8. Example 3. is a retraction then for Theorem If every topological space, every map is homotopic to the constant map Proof Define the homotopy by Question 5. Show that if X is any (n+1)-simplex and Y is its boundary and r : X  Y is a retraction and if Z is any topological space then every map f : Y  Z is homotopic to a constant map Remark In fact such a retraction does not exist. We will prove this fact in the last slide in this lecture in the case n = 2 and for n > 2 later.

9. Example 4. http://en.wikipedia.org/wiki/Real_projective_plane#Embedding_into_4-dimensional_space http://en.wikipedia.org/wiki/Rotation_group The rotation group SO(3) is topologically homeomorphic to the projective space Which can be identified with: i. set of lines through origin in http://en.wikipedia.org/wiki/Real_projective_space ii. quotient topology onwith antipodal points on its boundaryidentified (glued together)

10. Example 4. http://www.math.utah.edu/~palais/links.html Define maps d, but not s, is homotopic to a constant map by Proof Visual proof using a belt due to Dirac http://gregegan.customer.netspace.net.au/APPLETS/21/21.html Question 6. Construct a homotopy from d to a constant map

11. Covering Maps A map e : Y  X is a Definition Let X and Y be topological spaces and let D with and a homeomorphism there exists with open Example be a set with the discrete toplogy. covering map with fiber D if for every Proposition 4. If e : Y  X is a covering map and if g : [0,1]  X is a map and if p in Y with e(p) = g(0) then there exists a unique G : [0,1]  Y such that G(0) = p and e(G(y)) = g(y), y in Y. (G is called a lift of g) Question 7 Prove this result

12. Winding Number by Definition Let, choose by Let the unique lift of with, define define such that Question 8 Prove these results and define Proposition 5. W(f) does not depend on p. W(f) in Z Ifthen Corollary 2 Fundamental Theorem of Algebra Corollary 3 Brouwer’s Fixed Point Theorem for Question 10 State and prove this result Question 9 State and prove this result