MATH 7520: Algebraic Topology Homework
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MATH 7520 Algebraic Topology

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Fall 2007

### Homework Problems

- Show that a Δ-complex is a CW-complex.

- If
*X* is the torus, the real projective plane, the Klein bottle, your favorite other surface, exhibit a Δ-complex structure on *X*, and use it to compute the simplicial homology of *X*.

- Problem #8 from §2.1 in Hatcher

- Problem #23 from §2.1 in Hatcher

- Problems #11 - 13 from §2.1 in Hatcher

- Problems #16, 17, 20, 22 from §2.1 in Hatcher

- Compute the local homology groups
*H*_{*}(CT,CT-x), where *T* is the 2-dimensional torus, *CT* is the cone on *T*, and *x* is the cone point.

- Look at problems #1 - 8 from §2.2 in Hatcher

- Compute
*H*_{*}( **RP**^{n} ; **Z**/k* )*.

- Compute the homology and cohomology of the configuration space of 3 ordered points in the plane.

- Problems #6, 8, 9 from §3.1 in Hatcher

- Determine the structure of the cohomology rings
*H*^{*}( **RP**^{3} ; **Z**/2* )* and *H*^{*}( K ; **Z**/2* )*, where *K* is the Klein bottle.

- Problem #1 from §3.2 in Hatcher

*Dan Cohen
Fall 2007*

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