Math 872 - Section 001 - Spring 2010 - Problem sets

Instructions:
1) Problems marked with a * are to be handed in for grading on the due date for that problem set.
2) You may not use homework solutions written by me or any other faculty member (nor from the web) nor use verbal communications of those solutions from students who have these solutions or took the course in previous years, in the course of doing your homework.

• Problem set 1, due Tuesday 02/03:

• *(1.A.1) Write the connected sum of the torus T2 and the projective plane P2 either as a connected sum of tori or as a connected sum of projective planes. (Prove your answer with pictures!)

• (2.A.1) Define the loops f,g : I -> S2 based at (1,0,0) in S2 (with the Euclidean subspace topology) by f(s) := (cos(2 π s), sin(2 π s), 0) and g(s) := (1,0,0) for all s in I. Prove that f is path homotopic to g.

• (2.A.2) Hatcher p. 38 # 1

• (2.A.3) Let X := {a,b} have the indiscrete topology. Compute the fundamental group π1(X,a). (Prove your answer.)

• *(2.A.4) Hatcher p. 38 # 3

• *(2.A.5) Hatcher p. 38 # 5

• (2.B.1) (a) Show that for any continuous function h: (X,x0) -> (Y,y0), the induced map h*: π1(X,x0) -> π1(Y,y0) is a well-defined group homomorphism.
(b) Show that (1(X,x0))* = 1π1(X,x0).
(c) Let f: X -> Y be a continuous map, let x0,x1 ∈ X, and let α be a path in X from x0 to x1. Let f*,x0: π1(X,x0) -> π1(Y,f(x0)) and f*,x1: π1(X,x1) -> π1(Y,f(x1)) be the maps induced by f at x0 and x1 respectively, and let βα and βf ο α be the change of basepoint maps induced by the paths α in X and f ο α in Y, respectively. Prove that βf ο α ο f*,x0 = f*,x1 ο βα. (This part is Hatcher's problem p. 39 # 15.)

• Problem set 2, due Thursday, Feb 25:

• *(2.C) Show that every homomorphism φ : Z -> π1(X,x0) can be realized as the induced homomorphism φ = h* of a continuous map h : (S1, (1,0)) -> (X,x0).

• (2.C) Let p : R × R -> S1 × S1 be defined by p(x,y) := ( ( cos(2 π x) , sin(2 π x) ) , ( cos(2 π y) , sin(2 π y) ) ), and let f be the loop in S1 × S1 based at ((1,0),(1,0)) defined by f(s) := ( ( cos(2 π s) , sin(2 π s) ) , ( cos(4 π s) , sin(4 π s) ) ). Find a path g: (I,0) -> (R × R,(0,0)) satisfying p ο g = f. Sketch the image of g in R × R and the image of f in S1 × S1 (viewing S1 × S1 as the surface of a doughnut).

• (2.C.7) Hatcher p. 38 # 8

• *(2.C.7) Hatcher p. 38 # 9

• *(2.C.4) Hatcher p. 39 # 13

• *(2.C.?) Hatcher p. 39 # 16b (Hint: consider induced maps on π1).

• (2.C.4) Consider the map h: S1 → S1 given by h(z)=1/(zn) (where z ∈ S1 is considered as a complex number). What group homomorphism : ZZ is induced by h? (Prove your answer.)

• *(3.B.1) Let B := < a,b | aba = bab > (this is the braid group on 3 strands). Let H := < x | >, K := < y | >, and L := < z | > with group homomorphisms c : L -> H defined by c(z) := x2 and d : L -> K defined by d(z) := y3. Let G be the free product of H and K amalgamated along L via the maps c and d; that is, G = H *L K. Prove that the groups B and G are isomorphic.

Page written by C. Bleak; last updated 2/12/2010.