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\title{Homework 1: Due Thursday 1/17/19\\
MTH 961 W19}
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Note: For this homework you can assume that $\pi_nS^n\cong \bZ $ for $n\ge 1$, $\pi_{n+1}S^n\cong \bZ /2\bZ $ for $n\ge 3$, and $\pi_kS^n\cong 0$ for $k0$. Why does this not contradict the previous problem?
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\item We define
\[\bR P^n \co= \bR^{n+1}/\sim\]
where the equivalence relation $\sim$ identifies $x\in \bR^{n+1}$ with $\lambda \cdot x\in \bR^{n+1}$ for all $\lambda \in \bR$. In other words, this is the set real lines in $\bR^{n+1}$. Similarly,
define
\[\bC P^n \co= \bC^{n+1}/\sim \]
where for all $x\in \bC^{n+1}$ and all $\lambda\in \bC$ we have $x\simeq \lambda\cdot x$. This is the set of complex lines in $\bC^{n+1}$. Define a map
\[ S^n \to \bR P^n \]
by letting $S^n$ be the units in $\bR^n$ and sending $x\in S^n$ to $[x]\in \bR P^n$ and similarly define a map
\[ S^{2n+1} \to \bC P^n \]
by letting $S^{2n+1}$ be the units in $\bC^{n+1}$ and sending $x\in S^{2n+1}$ to $[x]\in \bC P^n$. When $n=1$ this recovers the first two Hopf fibrations from the problem before because $\bC P^1$ is homeomorphic to $S^2$ and $\bR P^1$ is homeomorphic to $S^1$. In general, this produces fiber sequences
\[ S^0 \to S^n \to \bR P^n \]
and
\[ S^1 \to S^{2n+1} \to \bC P^n \]
for $n\ge 1$.
Use these fiber sequences to compute $\pi_k\bR P^n$ for $k=0,1$ and $\pi_k\bR P^n$ for $k\ge 2$ in terms of the homotopy groups of spheres. Similarly, use these fiber sequences to compute $\pi_k\bC P^n$ for $0\le k \le 2$ and then compute $\pi_k\bC P^n$ in terms of homotopy groups of spheres for $k>2$.
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\item Given a sequence of CW complexes $X_i$ and maps $f_i\co X_i \hookrightarrow X_{i+1}$ for $i\ge 1$ where each $f_i$ is an inclusion of a sub-CW complex and $\cup_i X_i = X$, then
\[ \underset{ i }{\text{colim }} \pi_n X_i \cong \pi_n X. \]
For example, $\bR P^n\hookrightarrow\bR P^{n+1}$ and $\bC P^n\hookrightarrow \bC P^{n+1}$ are inclusions of sub-CW complexes. Use these sequences and the computations above to compute all of the homotopy groups of $\bR P^{\infty}=\cup_n\bR P^n$ and $\bC P^{\infty}=\cup_n\bC P^n$. (Prove that $\bR P^{\infty}$ is a $K(\bZ /2,1)$-space and $\bC P^{\infty}$ is a $K(\bZ ,2)$-space.)
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\item Prove that $X\times Y\to X$ is a fibration for any spaces $X$ and $Y$.
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\item Let $f\co X \to Z$ and $g\co Y \to Z$ be continuous maps and define the pullback (or fiber product) $X\times_Z Y$ to be the subset of $X\times Y$ given by
\[ X\times_Z Y = \{ (x,y) \mid f(x)=g(y) \} \]
equipped with the subspace topology. This space has the property that for any commutative diagram
\[
\xymatrix{
W \ar[ddr]_m \ar[drr]^n \ar@{-->}[dr]^(.6)h & & \\
& X\times_Z Y \ar[d] \ar[r] & Y \ar[d]^g \\
& X \ar[r]^f & Z
}
\]
the dotted arrow map exists; i.e. for any maps $m\co W\to X$ and $n \co W \to Y$ such that $f\circ m = g\circ n$, then the map $h\co W\to X\times_Z Y$ exists. This is the universal property of the pullback. Prove that if $p\co E\to B$ is a fibration and $g\co X\to B$ is any map then the map $X\times_{B}E\to X$ is also a fibration.
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\item Since we have an adjunction $\xymatrix{\Sigma \co \text{Top}_* \ar@<.5ex>[r] & \ar@<.5ex>[l] \text{Top}_* \rco \Omega}$, there is a unit map $X\to \Omega \Sigma X$ corresponding to $1_{\Sigma X}$ under the homeomorphism
\[ \text{Map}_*(\Sigma X,\Sigma X) \cong \text{Map}_*(X,\Omega \Sigma X).\]
Prove that the unit map $X\to \Omega \Sigma X$ induces a bijection after applying the functor $\pi_k$ for $k<2n-1$ and a surjection on $\pi_k$ for $k=2n-1$ when $X$ is $n$-connected.
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\item Read about the proof that $\pi_1^sS^0\cong \Omega_k^{\text{fr}}\cong \bZ /2\bZ $ using stably framed manifolds. A classical reference is the paper of Pontryagin ``Smooth manifolds and their applications in homotopy theory,'' but feel free to find your own reference. We will give a different proof of this fact using homotopy theory.
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