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INIDEPENDENT UNIVERSITY OF MOSCOW

4. COMPLEXES AND CELL HOMOLOGY

TOPOLOGY-2, FALL 2012

Problem 1 (5-lemma). Consider a commutative diagram
p

A


f -

A B C D E The lines of the diagram are exact, q and s are isomorphisms, p is an epimorphism, and t is a monomorphism. Prove that r is an isomorphism.

f -

q

B


g - g -

r

C


h - h -

s

i D - E


i -

t



is an exact sequence of complexes with the di erentials @A , @B and @ = 0. The map q : Bi Ci is onto, so take b Bi such that q(b) = c. = 0; hence, there exists Ai-1 such that p( ) = . (a) Prove that [ ] Hi-1 (A) is well-de ned, that is, does not depend on the choice of b (which is not unique). (c) Prove that [ ] depends only on [c] Hi (C ); this de nes a p q map i : Hi (C ) Hi-1 (A). (d) Prove that the sequence · · · Hi (A) - Hi (B ) - Hi (C ) - Hi-1 (A) : : : is exact. Problem 3. Let A -p B -q C 0 be an exact sequence of Abelian groups. Prove that for any Abelian group pid G the sequence A G - B G qid C G 0 is exact. Show that the similar statement for an exact sequence - 0 A -p B -q C may be not true.


Problem 2 (Bockstein's construction). Let 0

p q A - B - C 0 C , respectively. Let c Ci be such that @C c Denote = @B b Bi-1 ; then q( ) = p(@C c) @A = 0. (b) Prove that the homology class

@i+1 @i-1 @i @1 - Ci - Ci-1 - : : : - C0 - 0 be a @i+1 @i-1 @ @ a sequence · · · - Ci i- Ci-1 - · · · 1 C0 - 0 where Ci - i Gi where Gi is nite, complex. (b) Prove that if Hi (C ) = Z Show that Gi and Gi may be not isomorphic.

Problem 4. Let : : :


chain complex of free Abelian groups. Consider = Hom(Ci ; Z). (a) Prove that this is a cochain then H i (C ) = Z i Gi where Gi is nite, too.

Problem 5. For the following spaces nd CW-complexes homeomorphic to them and compute their homology
with the coe cients in Z and Z=2Z. If two spaces X and Y and a map f : X Y are given then do this for both spaces and compute the homomorphism f : H (X ) H (Y ). (a) X is a sphere with g handles; (b) X is the Klein bottle with g handles; (c) X is RP 2 with g handles; (d) X is the sphere with g handles and n holes; Y is the sphere with g handles, f : X Y is the natural embedding. (e) X = S n , Y = RP n , f : X Y is the universal cover. (f ) X = S 2n+1 = {(z0 ; : : : ; zn ) Cn | |z0 |2 + · · · + |zn |2 = 1}, Y = CP n , f : S 2n+1 CP n is f (z0 ; : : : ; zn ) = [z0 : · · · : zn ] (the generalized Hopf bundle). (g) X = S 3 = {(z ; w) C2 | |z |2 + |w|2 = 1}; the group Z=pZ = { m def e2im=p | m = 0; : : : ; p - 1} is acting on X by the maps m (z ; w) = ( m z ; qm w) where p and = q are coprime; Y is the orbit space for this action (called the (p; q)-lens, L(p; q)), and f : X Y is the map sending every point to its orbit. (h) X = S is the set of sequences (x1 ; : : : ; xn ; : : : ), such that in every sequence there are nitely many nonzero elements, and the sum of their squares is 1; Y = CP (what is it?), f : S CP is the ini nity-dimensional analog of the Hopf bundle. (i) Y = G(2; 4; R) (the set of 2-dimensional subspaces in R4 ), X = G+ (2; 4; R) (the set of oriented 2-dimensional subspaces in R4 ), f : X Y is forgetting the orientation.