### Math 430 homework, Fall 1999

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- Due Wed 1 Sep:

- (1.1)
- Show (for U open in the plane, f a real-valued function on U) that
f is locally constant iff f is constant on each component.
(See section A2, but note an error in the sentence before A.14.)
- (1.9)
- Show (for U open in the plane) that U is connected iff there is a
(piecewise smooth) path between any two points in U.
- (1.20)
- Let P and Q be points in the plane, and w
_{P} and w_{Q}
be the 1-forms corresponding to angular measure around these points.
Show that w_{P} - w_{Q} is exact.
What is a function f such that this form is df? (Ignore the hint in the
book!)

- Due Wed 8 Sep:

- 2.15, 2.20, 2.26

- Due Wed 15 Sep:

- 3.3, 3.14(i,ii), 3.17(a,b), 3.21

- Due Wed 22 Sep:

- 4.7, 4.15, 4.24, 4.28

- Due Wed 29 Sep:

- 5.4, 5.8, 5.16

- Due Wed 6 Oct:

- 6.5, 6.12, 6.14

- Due Wed 20 Oct:

- 10.7, 10.13, 10.14

- Due Fri 22 Oct:

- Look at
- A.3, 11.4, 11.10, 11.12
- Do
- 11.14, 11.21

- Due Wed 27 Oct:

- 12.5, 12.9, 12.14

- Due Wed 3 Nov:

- 13.2, 13.7, 13.14, 13.18

- Due Wed 10 Nov:

- read the ZIP classification of surfaces

- Due Wed 17 Nov:

- Look at
- 14.6, 14.8 (to prepare for today's test)

- Due Wed 8 Dec:

- If p:Y->X is a covering, then the induced map on fundamental
groups is injective. How about the induced map on H
_{1}?
Prove or give a counter-example.
- Prove the strong form of the Five Lemma (see Bredon 5.10).
If f
_{2} and f_{4} are surjective
and f_{5} in injective, then f_{3} is surjective.
If f_{2} and f_{4} are injective
and f_{5} in surjective, then f_{3} is injective.
- If A is a nonempty subset of X which is acyclic (has 0 reduced
homology in all dimensions) then show the relative homology of (X,A)
is the same as the reduced homology of X.