Indian Statistical Institute (ISI) 2006 M.Sc Mathematics Algebraic Topology - Question Paper

Attempt any 4 ques.. every ques. carries 25 marks. You may consult books and notes.

1. (i): Let C1 ,! C0 and D1 ,! D0 be 4 abelian groups. Show that:
(C0=C1) (D0=D1) '
C0 D0
C1 D0 + C0 D1
(Hint: View C: := 0 ! 0:: ! C1 ! C0 and D: := 0 ! 0::: ! D1 ! D0 as two-term chain complexes
and apply Kunneth formula to the tensor product chain complex C: D:)

(ii): Show that RP(2) is not a retract of RP(3).

2. (i): Prove that S1 is not a covering space of the bouquet of two circles S1 _ S1.

(ii): calculate homZ(Q;Q).

3. (i): Let f : T2 ! T2 be a continuous map, where T2 = S1_S1. Prove that f_ : H1(T2; Z) ! H1(T2; Z)
is an isomorphism i_ f_ : H2(T2; Z) ! H2(T2; Z) is an isomorphism. (Hint: Use the Z-cohomology
ring of T2.)

(ii): Let f : S2 ! T2 be a continuous map. Show that f_ : Hi(S2; Z) ! Hi(T2; Z) is the zero
homomorphism for i = 1; 2.

4. (i): Let fn : S1 ! S1 be the map z ! zn (n two N). Show that the topological mapping cone C(fn) is
homotopically equivalent to C(fm) i_ m = n.

(ii): Let M be a compact connected orientable manifold of dimension n. Let _ two Hi(M; Z) (where
0 _ i _ n) be a cohomology class such that _ [ _ = 0 for all _ two Hn_i(M; Z). Show that _ = 0.

5. (i): calculate H3(S2 _ RP(2); Z).

(ii): Prove that the map f : RP(2) ! RP(2) de_ned by [x0 : x1 : x2] 7! [x0 + 2x1 _ x2 : x1 _ 3x2 : x2]
does not lift to a map e f : RP(2) ! S2, where _ : S2 ! RP(2) is the usual covering projection.
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Earning:   Approval pending.