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Assigned Thursday March 24; due Thursday April 7, 3PM, in class

this document in PDF: HW.pdf

**Required reading. ** Read, reread and rereread
your notes to this point, and make sure that you really, really really,
really really really understand everything in them. Do the same every
week! Also, read Hatcher's pages 166-176 and 185-217.

**Solve the following problems. ** (But submit
only the underlined ones). In Hatcher's book, problems 1, __2__, 3, 4,
__5__, 10, __11__ on pages 176-177 ( and
are
defined on pages 6-7).

**Just for fun. ** Let be a nicely
embedded circle in
(``nicely'' means smoothly or even piecewise
polygonally with finitely many pieces, if you wish). We know that
.

- Understand this fact straight from the definition of homology on a pictorial level, without referring to any theorems whose proofs you cannot hold in your head in one piece.
- Now let be another nicely embedded circle in , disjoint from (so the two circles together form a 2-component link). Then represents a class in and hence an integer called . Use your understanding from the previous part to give a simple method to compute on a pictorial level.
- Prove that .
- How is related to the ``linking number'' defined in class a while ago, using the degree of a map ?

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Dror Bar-Natan 2005-03-23