## 8. Cellular homology

Use cellular chain complexes to compute the homology of the following spaces 1. $S ^ { m } \lor S ^ { n }$ and $S ^ { m } \times S ^ { n }$

2. The space obtained by identifying the edges of a solid hexagon as shown below.

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3. The orientable surface of genus g; for g = 2 the surface and a gluing pattern for it are shown.

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4. The non-orientable surface of genus h, made by removing h discs from a sphere and gluing M¨obius strips in their places.

5. The figure-of eight space $S ^ { 1 } \backslash S ^ { 1 }$ with two discs attached along the loops ababab and abababababab.

6. The space obtained by gluing up the sides of a solid tetrahedron as shown below (ABC is glued to ABD and ACD to BCD).

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7. The solid triangular prism with its faces identified as shown:

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8. The cube with faces identified in pairs according to the scheme below.

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9. The lens space $L ( p , 1 )$ (for p a natural number) is defined by the following gluing construction: take a closed lens-shaped 3-ball, partition its top and bottom hemispheres into p segments, and glue these together by vertical “translation” composed with a rotation by $2 \pi / p$ . The picture below shows the case $p = 6$ , with the lens drawn as a solid double hexagonal pyramid, and only three of the six pairs of glued faces are labelled, to avoid cluttering it up.

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Describe a CW complex structure for $L ( p , 1 )$ , and use this to compute its fundamental group and homology groups.

## Degree

10. Let p be a polynomial of algebraic degree n. Viewed as a continuous self-map of the Riemann sphere $S ^ { 2 } = \mathbb { C } \cup \{ \infty \}$ (sending infinity to infinity), what is its topological degree?

11. Consider the rational function $\begin{array} { r } { p ( z ) = \frac { ( z - a _ { 1 } ) ( z - a _ { 2 } ) \cdots ( z - a _ { n } ) } { ( z - b _ { 1 } ) ( z - b _ { 2 } ) \cdots ( z - b _ { m } ) } } \end{array}$ , where all the $a _ { i }$ and $b _ { j }$ are distinct from one another, as a function from the Riemann sphere to itself.
    What is its topological degree?

12. Let $f$ be a map $S ^ { m } \to S ^ { n }$ for which there exists some open set U such that $f | _ { U }$ is a homeomorphism to its image.
    Show that $m = n$ and f is surjective.