Show that the inclusion of the real projective plane in the complex projective plane is not null-homotopic?

Question

In other words, how to show that $\mathbb{RP}^2$ is not contractible in $\mathbb{CP}^2$.

Any hints would be appreciated.

Answer 1

I will accept Georges' challenge.

1) Cohomological. The fundamental class of the real projective plane defines a class $[\Bbb{RP}^2] \in H_2(\Bbb{CP}^2;\Bbb Z/2)$. To see that it is non-trivial, recall that the cup product in cohomology is Poincare dual to the intersection product in homology. If we can show that $[\Bbb{RP}^2] \frown [\Bbb{RP}^2] = [pt]$, then we have what we want. Take one of these classes to be represented by $\{[x_0 : x_1 : x_2] \mid x_i \in \Bbb R\}$ and another to be $\{[x_0 : \omega x_1 : \omega^2 x_2] \mid x_i \in \Bbb R\}$ where $\omega$ is your favorite nontrivial cube root of unity. (These two embeddings are homotopic, hence homologous.) For $[x_0 : x_1 : x_2] = [y_0 : \omega y_1 : \omega^2 y_2]$, where all $x_i$ and $y_i$ are real, we must have $x_1 = y_1 = x_2 = y_2 = 0$, and hence the only intersection point is $[1:0:0]$. The two tangent spaces of these embeddings are $\Bbb R^2 \subset \Bbb C^2$ and $\Bbb R \omega \oplus \Bbb R \omega^2 \subset \Bbb C^2$, which sum to the whole $\Bbb C^2$, so this is a transverse intersection, and the product is indeed one point (and hence nontrivial, as desired).

2) An immediate route to the above computation comes from symplectic geometry: $\Bbb{RP}^2$ is Lagrangian and hence the normal bundle is isomorphic to its tangent bundle; so the self-intersection number is $\chi(\Bbb{RP}^2) = 1$.

3) Or observe that the restriction of the tautological line bundle of $\Bbb{CP}^2$ is the complexification of the tautological line bundle on $\Bbb{RP}^2$, and hence has $c_1(\ell_{\Bbb C}) = \beta w_1(\ell_{\Bbb R}) \neq 0 \in H^2(\Bbb{RP}^2;\Bbb Z/2)$. Here we use either the general formula for a real vector bundle, $c_{2i+1}(E_{\Bbb C}) = \beta w_{4i+1}(E)$, or think of the Bockstein $\beta$ as the composite $H^1(X;\Bbb Z/2) \to H^1(X; U(1)) \to H^2(X;\Bbb Z)$, sending a flat real line bundle to a flat complex line bundle with the same transition functions in $\pm 1 \subset U(1)$ and then forgetting the flat structure.

4) Amusingly, because $\pi_2(\Bbb{CP}^2) \to H_2(\Bbb{CP}^2)$ is an isomorphism, we see that we may compute $\pi_2(\Bbb{RP}^2) \to \pi_2(\Bbb{CP}^2)$ as the map $\pi_2(\Bbb{RP}^2) \to H_2(\Bbb{RP}^2) \to H_2(\Bbb{CP}^2)$; of course, $H_2(\Bbb{RP}^2;\Bbb Z) = 0$, so the map is not detected on $\pi_2$. Further, by considering the diagram comparing the fiber sequences $S^1 \to S^5 \to \Bbb{CP}^2$ and $O(2) \to S^3 \to \Bbb{RP}^2$ and the fact that the inclusion $S^3 \to S^5$ is null-homotopic, we see that the map is zero on all homotopy groups.

There is still something the homotopy theory can say, though. If $\Bbb{RP}^2 \to \Bbb{CP}^2$ was null-homotopic, there would in particular be a lift of the initial map to a map to $S^5$ (using the homotopy lifting lemma), and in particular by restriction a section $\Bbb{RP}^2 \to S^3$ of the original fibration. There are a number of reasons this is impossible: one is that there are no embedded $\Bbb{RP}^2$s in $S^3$.

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On the relation $c_1(\ell \otimes \Bbb C) = \beta w_1(\ell)$ for real line bundles $\ell$.

We may define a map $H^1(X; \Bbb Z/2) \to H^1(X; U(1)) \to H^2(X;\Bbb Z)$, the first by inclusion of coefficients $\pm 1 \to U(1)$ and the second by the boundary map in the long exact sequence $H^1(X; \Bbb R) \to H^1(X; U(1)) \to H^2(X; \Bbb Z)$. In terms of calculability, that long exact sequence helps you identify $H^1(X; U(1))$ as an extension of $H^2_{\text{tors}}(X; \Bbb Z)$ by $U(1)^{b_1}$, and the boundary map is just projection to the group of components $H^2_{\text{tors}}(X;\Bbb Z)$, which then includes into $H^2(X;\Bbb Z)$. In any case, all you need to know to calculate the kernel of this map is which classes of $H^1(X; \Bbb Z/2)$ lift to classes of $H^1(X;\Bbb Z)$, and both of these have nice definitions in terms of homomorphisms out of the fundamental group.

At the level of cocycles, if $\sigma$ is a $\Bbb Z/2$-cocycle and $\tilde \sigma$ is a lift to a $\Bbb Z$-cochain, then just as we considered $\sigma$ as a $U(1)$-cocycle above we may consider $\tilde \sigma$ as a $\Bbb R$-cochain. The boundary map $H^1(X; U(1)) \to H^2(X; \Bbb Z)$ takes a lift of $\sigma$ as a $U(1)$-cocycle to a $\Bbb R$-cochain $\tilde \sigma$, and restricts $\partial \tilde \sigma$ to a $\Bbb Z$-valued cocycle. By definition, $\beta[\sigma] = [\partial \tilde \sigma]$, and so we see that the procedure outlined above gives another definition of the Bockstein.

Now interpreting the three groups in terms of line bundles, $H^1(X;\Bbb Z/2)$ is isomorphic to the group of real line bundles, $H^1(X; U(1))$ is isomorphic to the group of flat complex line bundles (those equipped with some system of local trivializations with constant transition maps), and $H^2(X; \Bbb Z)$ is the group of complex line bundles. Writing $\mathcal U(1)$ for the sheaf of continuous maps to $U(1)$ and similarly $\mathcal R$ for continuous maps to $\Bbb R$, the long exact sequence of cohomology for $\Bbb Z \to \mathcal R \to \mathcal U(1)$ identifies $H^1(X; \mathcal U(1)) = H^2(X; \Bbb Z)$, and comparing this sequence to that of $\Bbb Z \to \Bbb R \to U(1)$ identifies the map $H^1(X; U(1)) \to H^1(X; \mathcal U(1)) \cong H^2(X; \Bbb Z)$ as the boundary map we described at the beginning.

In particular, $H^1(X;\Bbb Z/2) \to H^1(X; U(1))$ sends a real line bundle, perhaps described as a collection of local trivializations with transition maps in $\Bbb Z/2$, to a complex line bundle with the same local trivializations and the same transition maps $\Bbb Z/2 \subset U(1)$; we consider this as a flat line bundle. (This is sending $\ell$ to $\ell \otimes \Bbb C$ equipped with a particular flat structure induced by that of $\ell$.) The above paragraph identifies the next map $H^1(X; U(1)) \to H^2(X; \Bbb Z)$ as forgetting the flat structure. Thus the composite $H^1(X; \Bbb Z/2) \to H^2(X;\Bbb Z)$ sends $\ell \mapsto \ell \otimes \Bbb C$, considered as real and complex line bundles, respectively. Because $w_1$ and $c_1$ give the isomorphisms between groups of line bundles and these cohomology groups, and we identified that composite above as the Bockstein $\beta$, we have seen what we wanted.