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I'm having a little trouble understanding several topics from algebraic topology. This question covers a range of topics I have been looking at.

Can anyone help? Thanks!

Suppose $X$ and $Y$ are connected manifolds, $X$ is simply connected, and the universal cover of $Y$ is contractible. Why is every continuous mapping from $X$ to $Y$ homotopic to a constant?

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First observe that if $g: X \to Y$ is a map with $Y$ contractible then $g$ is homotopic to a constant map. For let $h_t: Y\to Y$ be such that $h_t(y) = y, h_1(y) = y_0$. Then $g\circ h_t$ is a homotopy from $g$ to the constant map $y \mapsto g(y_0)$. Similarly, if we precompose a map that is homotopic to a constant with another map, then the composition is also homotopic to a constant.

Now let $p:\tilde Y \to Y$ be a cover of $Y$. A basic fact about covering spaces is that a map $f: X \to Y$ lifts to a map $\tilde f : X \to \tilde Y$ if $f_* \pi_1(X) \subset p_* \pi_1 (\tilde Y)$. So if $X$ is simply connected, this is always true. So in your case, taking $\tilde Y$ to be the universal cover, any map $f: X\to Y$ lifts to a map $\tilde f: X \to \tilde Y$ and $\tilde f$ must be homotopic to a constant map since $\tilde Y$ is contractible. Therefore $f = p \circ \tilde f$ is also homotopic to a constant map.

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Let $\tilde{Y} \xrightarrow{\pi} Y$ be the universal cover of $Y$. Since $X$ is simply connected, any continuous map $X \xrightarrow{f} Y$ can be factorized as a continuous map $X \xrightarrow{\tilde{f}} \tilde{Y} \xrightarrow{\pi} Y$. Since $\tilde{Y}$ is contractible, there is a point $y \in \tilde{Y}$ and an homotopy $h$ between the identity map on $\tilde{Y}$ and the constant map $y$ :

$h : \begin{array}{c}\tilde{Y} \xrightarrow{id} \tilde{Y} \\ \Downarrow \\ \tilde{Y} \xrightarrow{y} \{y\}\end{array}$

Composing this homotopy with $\tilde{f}$ and $\pi$, you get an homothopy $h'(t,x) = \pi(h(t,\tilde{f}(x))$

$h': \begin{array}{rcl}X \xrightarrow{\tilde{f}} & \tilde{Y} \xrightarrow{id} \tilde{Y} &\xrightarrow{\pi} Y \\ &\Downarrow &\\ X \xrightarrow{\tilde{f}} & \tilde{Y} \xrightarrow{y} \{y\} & \xrightarrow{\pi} \{\pi(y)\} \end{array}$ between $f$ and the constant map $\pi(y)$.

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