Algebraic topology qualifying exam February 2000
Answer eight questions, four from part I and four from part II. Give as much detail in
your answers as you can.
Part I
1. a) Let Gand Hbe functors from a category Cto a category D. Define a natural
transformation from Gto H. b) For an admissible pair of topological spaces (X,A)
define functors Gand Hby G(X,A)=Hp(X,A), H(X,A)=Hp-1(X,A). Show that the map
∂*is a natural transformation of Gto H. Define and give an example (with proof) of a
contravariant functor.
2. State and prove the Kunneth theorem for topological spaces.
3. a) Let Fbe the closed orientable surface of genus 2. Find a presentation for the
fundamental group of F. b) Prove that πn(X) is abelian for n>1.
4. State and prove Poincaré duality for orientable triangulated compact homology n-
manifolds.
5. a) Give the definition of cap products (∩).
b) Show that, for Ka simplicial complex and coefficients R,itgivesa
homomorphism Hp(K,R)⊗Hp+q(K,R)→Hq(K,R).
Part II
6. The cyclic group G=Z/pZ (pa prime) acts on the unit sphere S2n-1 as follows: let z=
exp(2πi/p)beapth root of unity and think of Gas the subgroup of the complex
numbers generated by zunder multiplication. Then the action is: z(x1,…,x2n)=
(zx1,…,zx2n). Let Ln=S2n-1/(Z/pZ). Why is Lna manifold? Find H*(Ln,Z)(whereZ=
integer ring).
7. a) Find H*(Sn×Sm). b) Find H*(Sn∨Sm∨Sn+m). What do you notice?
8. Let Mbe a connected compact n-manifold with boundary Bwhere n> 1. a) show that
Bis not a retract of M. b) Prove that if Mis contractible, then Bhas the homology of a
sphere.
9. Let T=S1×S1×S3. Find the cohomology groups of Tand its ring structure with
coefficients the integers.
10. a) What are (Z/mZ)*Zand Ext(Z/mZ,Z). Here * denotes the torsion product and you
must give a proof in each case. Zdenotes the integers.
b) Find H*(S1×RP2;Z/mZ).