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MATH 7520 Algebraic Topology Fall 2002

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Homework Assignments - to be turned in

- Verify property (2) of simplices:

The simplex spanned by *a*_{0},...,*a*_{n} is equal to the union of
all line segments joining *a*_{0} to points of the
simplex spanned by *a*_{1},...,*a*_{n}.

Two such line segments intersect only in the point *a*_{0}.

#4, section 1, page 7

- #5, section 2, page 14

#2 - 6, section 5, page 33

- #2, section 6, page 40

#1, section 8, page 46

#3, 6, section 9, page 51

- #2 (a) and (b), section 14, page 83

#1, section 16, page 95

- #1, section 21, page 120

#1, 2, section 22, page 127

- #4, section 23, page 136

#6 (a) - (c), section 24, page 141

#1, section 25, page 144

- Compute the singular homology of the "pseudo-projective space"
**X**_{n} = **S**^{1} U_{fn}
**D**^{2}.

Here **S**^{1} is the unit circle, the set of all complex numbers
**z** of length 1;

**D**^{2} is the disk, the set of all complex
numbers of length at most 1;

f_{n}:**S**^{1} ->
**S**^{1} is given by
f_{n}(**z**) = **z**^{n}; and

**X**_{n} is formed by identifying **z** in the boundary of
**D**^{2} with f_{n}(**z**) in **S**^{1}.

Note that **X**_{1} = **D**^{2} and
**X**_{2} = **RP**^{2}.

Let **M**_{g} = T # T # ... # T be the surface obtained by
forming the connected sum of g tori.

Compute the singular homology of
**M**_{g}.

If **N**_{h} = **RP**^{2} # **RP**^{2}
# ... # **RP**^{2} is the surface obtained by
forming the connected sum of h copies of the projective plane,

can you compute the singular homology of
**N**_{h}?

- Let Y be a k-cell in
**R**^{n}.
Determine H_{*}(**R**^{n} - Y).

Let A be a subset of **R**^{n}, homeomorphic
to the k-sphere **S**^{k} for k < n.
Determine H_{*}(**R**^{n} - A).

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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

read sections 10 - 12

read sections 14 - 17

#5, section 24

#3, section 25