Math 742- Topology
March 3, 2010
Homework no. 3
[Due February 22]
1. Recall that H 2 (RP2 ; Z2 ) Z2 with Z2 generated by . Find an explicit repre=
sentation of for the -complex structure on RP2 given in Figure 1. Specically,
since
=
2 : Coker : Hom(Z1 (RP
Math 742- Topology
February 10, 2010
Homework no. 2
[Due February 22]
1. Compute the cup product structure on H (#h RP2 ; Z2 ) where #h RP2 is the nonorientable surface gotten by taking the connected sum of h 2 copies of RP2 .
Note that the coecient group
Math 742- Topology
February 1, 2010
Homework no. 1
[Due February 10]
1. Given Abelian groups H1 , H2 , G, prove the isomorphisms:
a) Ext(H1 H2 , G) Ext(H1 , G) Ext(H2 , G).
=
b) Ext(H, G) = 0 for any free Abelian group H .
c) Ext(Zn , G) G/(nG).
=
2. Let
The cup product on H (2 ; Z)
a1
x
b2
x
x
Im(1 )
Im(8 )
c8
c1
c7
b1
Im(2 )
x
c2
y
a2
x
Im(7 )
c6
Im(3 )
c3
Im(6 )
c5
c4
x
a2
v1
=
2
R2
x
Im(4 )
Im(5 )
b2
a1
b1
x
v2
1
2
8
v8
v3
w
7
v7
3
6
4
v4
5
v6
v5
Figure 1. The -complex structure on the surface 2 .
1.
The cup product on H (RP2 ; Z2 ) revisited
a1
Im(1 )
y
c2
c1
=
x
RP2
x
R2
Im(2 )
a1
v1
v2
v6
1
2
v3
v5
v4
Figure 1. Another -complex structure on the surface RP2 .
1. The chain and homology groups of RP2
In each of the next 3 subsections, we list the n-si
The cup product on H (RP2 ; Z2 )
a1
y
x
Im(1 )
c1
c4
z
b1
Im(4 )
=
RP2
b1
R2
Im(2 )
c2
c3
Im(3 )
x
y
a1
v2
v1
1
4
w
2
3
v4
v3
Figure 1. The -complex structure on the surface RP2 .
1. The chain and homology groups of RP2
In each of the next 3 subsections,