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\large{2015 Spring Math 5043 \quad Assignment 1. \qquad Due: Jan 30, 2015}
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\large{Name: }
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1. Show that $f: X \to Y$ is a homotopy equivalence if there exits maps $g, h: Y \to X$ such that $fg$ and $hf$ are homotopy equivalences.

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2. Show that a cell complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible.

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3. Show that $S^1 \ast S^1 = S^3$, and more generally, $S^m \ast S^n = S^{m+n+1}$.

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Hint: Use the formal linear combination.

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4. (Not to be graded) Let $M$ be a connected smooth manifold. Show that the fundamental groupoid $\Pi_1(M)$ is also a connected smooth manifold. Moreover, show that the source and target maps
\begin{equation}
	s, t: \Pi_1(M) \to M
\end{equation}
are submersions.

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5. Hatcher 1.1.13
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Given $X$ and a path-conncted subspace $A$ containing the basepoint $x_0$, show that the map
\begin{equation}
	\pi_1(A, x_0) \to \pi_1(X, x_0)
\end{equation}
induced by the inclusion $A \hookrightarrow X$ is surjective if and only if every path in $X$ with basepoints in $A$ is homotopic to a path in $A$.
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6. Hatcher 1.1.16 (c), (d), (f)

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Fruit for thought, not to be graded:

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I. We give a different cell complex structure to $S^n$, where each $S^k$ as the equator of $S^{k+1}$
\begin{equation}
	S^0 \subset S^1 \subset \cdots \subset S^n
\end{equation}
is a subcomplex. In this cell complex structure, the $k$-skeleton is simply $S^k$, and $S^{k+1}$ is obtained by attaching two $k$-cells, the components of $S^{k+1} \setminus S^k$ to $S^k$. In this case, the infinite-dimensional sphere
\begin{equation}
	S^\infty = \bigcup_n S^n
\end{equation}
is also a cell complex. Now, show that $S^\infty$ is contractible.

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