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\large{2014 Fall Math 5041 \quad Assignment 4. \qquad Due: Nov 7, 2014}
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\large{Name: }
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1.
Let $M$ be a smooth manifold with boundary. Show that there exists a smooth function $f: M \to [0, \infty)$ such that $0$ is a regular value of $f$ and $\partial M = f^{-1}(0)$.
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2. a) For the zero vector field $X_0: \mathbb{R}P^2 \to T(\mathbb{R}P^2)$, compute the mod 2 self intersection number of $X_0$.
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3. Prove that intersection theory is vacuous in contractable manifolds: if $Y$ is contractable and $\dim Y > 0$, then $I_2(f, Z) = 0$ for every $f: X \to Y$, with $X$ and $Z$ being compact, and $\dim X + \dim Z = \dim Y$.
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Remark: In particular, intersection theory is vacuous in Euclidean space.
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4. For topological space $M$, we say $M$ is contractible if the identity map $\mathtt{id}: M \to M$ is homotopic to some constant map. Show that if $M$ is a compact manifold, which is not a single point, then $M$ is not contractible.
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5. Show that if $X$ is compact, but $Y$ is not compact, then $\mathtt{deg}_2(f) = 0$ for all smooth maps $f: X \to Y$. Of course, we assume that $\dim X = \dim Y$.
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6. Show that the sphere $S^2$ and the torus $T^2$ are not diffeomorphic.
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\textbf{Bonus:} Is there a smooth map $f: S^2 \to T^2$ such that $\mathtt{deg}_2(f) = 1$, and is there a smooth map $g: T^2 \to S^2$ such that $\mathtt{deg}_2(g) = 1$?
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