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\large{2014 Fall Math 5041 \quad Assignment 2. \qquad Due: Oct 29, 2014}
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\large{Name: }
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1. Show that Brouwer fixed point theorem is false if the closed ball $D^n = \overline{B^n}$ is replaced with the open ball $B^n$. That is, find a continuous map $f: B^n \to B^n$ with no fixed point.

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2. Let $M$ be the solid torus. For example, we may define it to be
$$
	M = \{((2 + r \cos \theta) \cos{\varphi}, (2 + r \cos \theta) \sin{\varphi}, r \sin \theta) \in \mathbb{R}^3 ~|~ \theta, \varphi \in \mathbb{R} / 2\pi \mathbb{Z}, 0 \leq r \leq 1 \}
$$

 Find a continuous map $f: M \to M$ such that $f$ has no fixed point.

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3. Let $A$ be a $n \times n$ matrix such that all entries of $A$ are non-negative. Show that $A$ has a non-negative eigenvalue.

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4. Let $X, Y$ be regular submanifolds of $\mathbb{R}^N$. Show that for almost all $a \in \mathbb{R}^N$, $X + a$ intersects $Y$ transversally.


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