2006 Indian Statistical Institute M.Sc Mathematics Algebraic Topology University Question paper for exam preparation. Question paper for 2006 Indian Statistical Institute M.Sc Mathematics Algebraic Topology University Question paper, Exam Question papers 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2012 university in india question papers. SiteMap
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2006 Indian Statistical Institute M.Sc Mathematics Algebraic Topology University Question paper

University Question Papers
2006 Indian Statistical Institute M.Sc Mathematics Algebraic Topology University Question paper
Attempt any four questions. Each question carries 25 marks. You may consult books and notes.

1. (i): Let C1 ,! C0 and D1 ,! D0 be four 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 2 circles S1 _ S1.

(ii): Compute 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 2 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 _ 2 Hi(M; Z) (where
0 _ i _ n) be a cohomology class such that _ [ _ = 0 for all _ 2 Hn_i(M; Z). Show that _ = 0.

5. (i): Compute 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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