$[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Question

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.

Answer 1

Here I claim that the correspondence $F:[M,\mathbb CP^2]\rightarrow[M,\mathbb CP^\infty]$ defined by sending $[f]$ to the homotopy class of $f$ as a map into $\mathbb CP^\infty$, is a bijection. If two maps are homotopic as maps into $\mathbb CP^2$, then so are they as maps into $\mathbb CP^\infty$ so the correspondence is well-defined.

It is clear that every map $f:M\rightarrow\mathbb CP^\infty$ can be homotoped to lie in the canonically included $\mathbb CP^2$ by the facts that every differential manifold has the homotopy type of a CW complex and that every map of CW complexes is homotopy to a cellular map, i.e., the k-skeleton maps into the k-skeleton for each k. So basically the map at time 1 in the homotopy has image in $\mathbb CP^1\subset\mathbb CP^2$. This shows that every homotopy class in $[M,\mathbb CP^\infty]$ is represented by a map into $\mathbb CP^2$, which shows that the correspondence $F$ is surjective.

For injectivity, I have to show that if $f,g:M\rightarrow\mathbb CP^2$ are homotopic as maps into $\mathbb CP^\infty$, then so are they as maps into $\mathbb CP^2$, or that a homotopy of such maps, as maps into $\mathbb CP^\infty$ descends to that as maps into $\mathbb CP^2$

Equivalently, I have to show that a map $F:M\times[0,1]\rightarrow\mathbb CP^\infty$ which sends $X=M\times\{0,1\}$ into $\mathbb CP^2$, is homotopic rel $X$ to a map $G$ with image in $\mathbb CP^2$.

$M\times [0,1]$ is a 4-d complex formed by attaching a 4-cell by mapping the boundary onto $M\times\{0,1\}\cup e_0\times[0,1]$ by the appropriate attaching map, where $e_0$ is a 0-cell. Now $M\times[0,1]\setminus X$ has the 1-cell $e_0\times(0,1)$ and the interior of the 4-cell attached. By the compression lemma 4.6 in Hatcher's Algebraic Topology book, it boils down to showing that $\pi_1(\mathbb CP^\infty,\mathbb CP^2)$ and $\pi_4(\mathbb CP^\infty,\mathbb CP^2)$ vanish.

By the homotopy exact sequence for a pair, $\pi_1(\mathbb CP^\infty,\mathbb CP^2)=0$ and $\pi_4(\mathbb CP^\infty,\mathbb CP^2)\cong\pi_3(\mathbb CP^2)$.

One then applies the homotopy exact sequence for $S^1\hookrightarrow S^5\rightarrow\mathbb CP^2$ and computes $\pi_3(\mathbb CP^2)=0$ $\square$