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TOPOLOGY II
ASSIGNMENT 4 (CELLULAR HOMOLOGY)
Problem 4.1. Compute H # (M 2
g ), where M 2
g is the sphere with g handles.
Problem 4.2. Compute H # (N 2
g ), where N 2
g is RP 2 with g handles.
Problem 4.3. Compute H # (CP n ).
Problem 4.4 # . Compute H # (RP n ).
Problem 4.5. Compute H # (#), where # is the following graph.
Problem 4.6. Let p and q be two relatively prime positive integers. Consider the action of the
group Z p with the generator # on the unit sphere S 3
# C 2 defined by
#(z, w) = (exp(
2#i
p
), exp(
2#iq
p
)).
The quotient of S 3 by this action is a 3­manifold. It is called a lens space and is denoted by
L(p, q). Compute H # (L(p, q)
Problem 4.7. Compute H # (D), where D is the dunce hat, i.e., the triangle with the identifi­
cations shown by arrows.
Problem 4.8. Compute H # ( •
S), where •
S is the sphere S 2 to which the segment [NS] joining
the North and South poles have been added.
Problem 4.9. Express H n (A # B) in terms of H n A and H n B, n = 0, 1, . . .
Problem 4.10. Is it possible to retract the dunce hat to its circle NM ?