## MATH 7520   Algebraic Topology   Fall 2002

### Homework Assignments - to be turned in

1. Verify property (2) of simplices:
The simplex spanned by a0,...,an is equal to the union of all line segments joining a0 to points of the simplex spanned by a1,...,an.
Two such line segments intersect only in the point a0.
#4, section 1, page 7

2. #5, section 2, page 14
#2 - 6, section 5, page 33

3. #2, section 6, page 40
#1, section 8, page 46
#3, 6, section 9, page 51

4. #2 (a) and (b), section 14, page 83
#1, section 16, page 95

5. #1, section 21, page 120
#1, 2, section 22, page 127

6. #4, section 23, page 136
#6 (a) - (c), section 24, page 141
#1, section 25, page 144

7. Compute the singular homology of the "pseudo-projective space" Xn = S1 Ufn D2.
Here S1 is the unit circle, the set of all complex numbers z of length 1;
D2 is the disk, the set of all complex numbers of length at most 1;
fn:S1 -> S1 is given by fn(z) = zn; and
Xn is formed by identifying z in the boundary of D2 with fn(z) in S1.
Note that X1 = D2 and X2 = RP2.

Let Mg = T # T # ... # T be the surface obtained by forming the connected sum of g tori.
Compute the singular homology of Mg.

If Nh = RP2 # RP2 # ... # RP2 is the surface obtained by forming the connected sum of h copies of the projective plane,
can you compute the singular homology of Nh?

8. Let Y be a k-cell in Rn.   Determine H*(Rn - Y).

Let A be a subset of Rn, homeomorphic to the k-sphere Sk for k < n.   Determine H*(Rn - A).

### Stuff to look at - not turned in

#3 - 5, section 6, page 40
#1, section 7, page 43
do one of the problems in section 12, page 70