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# 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 ($f(open)\ne open$).