### 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 wP and wQ be the 1-forms corresponding to angular measure around these points. Show that wP - wQ 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:
1. If p:Y->X is a covering, then the induced map on fundamental groups is injective. How about the induced map on H1? Prove or give a counter-example.
2. Prove the strong form of the Five Lemma (see Bredon 5.10). If f2 and f4 are surjective and f5 in injective, then f3 is surjective. If f2 and f4 are injective and f5 in surjective, then f3 is injective.
3. 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.