Topology II -- Spring 2014 -- midterm

Name:_________________________

10 problems, 10 points each

• Justify every step you make with as thorough explanation as possible.
• Unless requested, no decimal representation of the answers is necessary.
• Start every problem at the top of the page, in the given order.

$\bullet$ 1. Compute the homology of the cell complex that consists of $n$ circles attached to each other consecutively (chain-like).

$\bullet$ 2. Compute the homology of the cell complex that consists of the sphere with a string connecting the north and south poles.

$\bullet$ 3. Let $Q$ be the solid unit cube in ${\bf R}^3$ and let $X=Q / _{\sim}$ be the cube with the opposite faces identified.

• (a) Describe the equivalence relation.
• (b) Represent $X$ as a cell complex: cells, skeleta, attaching maps.

$\bullet$ 4. Describe and sketch a complex with the following homology groups:

• (a) $H_0={\bf Z},H_1={\bf Z},H_2={\bf Z}$.
• (b) $H_0={\bf Z},H_1={\bf Z}_2,H_2={\bf Z} \times {\bf Z} \times {\bf Z}$.

(No proof necessary.)

$\bullet$ 5. Is the preimage of a Hausdorff space under a continuous function always Hausdorff?

$\bullet$ 6. Give an example of a continuous function ($f^{-1}(open)=open$) that takes every closed set to a closed set ($f(closed)=closed$), but doesn't always take every open set to an open set ($f(open)\ne open$).