cohomology of complex projective space

hbghlyj

For $n>m>1$, we can consider $\mathbb{CP}^m$ as a subspace of $\mathbb{CP}^n$ by
\[
\mathbb{CP}^m=\left\{\left[z_0: \cdots: z_n\right] \mid z_{m+1}=\cdots=z_n=0\right\} .
\]
Let $M=\mathbb{CP}^n \setminus \mathbb{CP}^m$. Compute the following cohomology groups (with compact support) for $i \geq 0$,

• $H^i(\mathbb{CP}^n, \mathbb{CP}^m ; \mathbb{Z})$;
• $H_c^i(M ; \mathbb{Z})$;
• $H^i(M ; \mathbb{Z})$.

hbghlyj

• $\mathbb{CP}^n$ has a CW complex structure consisting of one cell in each even dimension: 0, 2, 4, ..., $2n$. Similarly, $\mathbb{CP}^m$ has cells in dimensions 0, 2, ..., $2m$. The pair $(\mathbb{CP}^n, \mathbb{CP}^m)$ thus has relative cells in dimensions $2(m+1)$, $2(m+2)$, ..., $2n$.
  
  The cellular cochain complex for the pair has groups $\mathbb{Z}$ in these even relative dimensions and zero elsewhere. Since there are no cells in odd dimensions, all differentials are zero. Therefore, the relative cohomology is the same as the cochain groups.
  
  This yields:
  \[
  H^i\left(\mathbb{CP}^n, \mathbb{CP}^m ; \mathbb{Z}\right) =
  \begin{cases}
  \mathbb{Z} & \text{if } i = 2k \text{ for } m+1 \leq k \leq n, \\
  0 & \text{otherwise}.
  \end{cases}
  \]
  Since $H^*(\mathbb{CP}^n ; \mathbb{Z}) \cong \mathbb{Z}[x]/(x^{n+1})$ and $H^*(\mathbb{CP}^m ; \mathbb{Z}) \cong \mathbb{Z}[y]/(y^{m+1})$ with $\deg x = \deg y = 2$, the cohomology ring is additively $\bigoplus_{k=m+1}^n \mathbb{Z}u_k$, where $\deg u_k = 2k$, and the cup product is given by $u_j \cup u_k = u_{j+k}$ if $j+k \leq n$, and $0$ otherwise.
• The compactly supported cohomology $H_c^i(M; \mathbb{Z})$ of a locally compact space $M$ can be defined as the relative cohomology $H^i(M^+, \infty; \mathbb{Z})$, where $M^+$ denotes the one-point compactification of $M$.
  In this case, $M = \mathbb{CP}^n \setminus \mathbb{CP}^m$ is the complement of the closed subspace $\mathbb{CP}^m$ in the compact space $\mathbb{CP}^n$. The one-point compactification $M^+$ is homeomorphic to the quotient space $\mathbb{CP}^n / \mathbb{CP}^m$, where the subspace $\mathbb{CP}^m$ is collapsed to a single point $p$ (which corresponds to $\infty$ in $M^+$). Thus, $H_c^i(M; \mathbb{Z}) \cong H^i(\mathbb{CP}^n / \mathbb{CP}^m, p; \mathbb{Z})$.
  Since $\mathbb{CP}^n$ and $\mathbb{CP}^m$ are CW complexes and $\mathbb{CP}^m$ is a subcomplex of $\mathbb{CP}^n$ (with the given cell structures), the pair $(\mathbb{CP}^n, \mathbb{CP}^m)$ is a CW pair. For such pairs, the quotient map $q: \mathbb{CP}^n \to \mathbb{CP}^n / \mathbb{CP}^m$ (which sends $\mathbb{CP}^m$ to $p$) induces an isomorphism $q^*: H^i(\mathbb{CP}^n / \mathbb{CP}^m, p; \mathbb{Z}) \to H^i(\mathbb{CP}^n, \mathbb{CP}^m; \mathbb{Z})$. Thus,
  \[
  H_c^i(M ; \mathbb{Z}) =
  \begin{cases}
  \mathbb{Z} & \text{if } i = 2k \text{ for } m+1 \leq k \leq n, \\
  0 & \text{otherwise}.
  \end{cases}
  \]
• Coordinatize $\mathbb{C}^{n+1} = W \oplus U$ where $\dim W = m+1$ (corresponding to $\mathbb{CP}^m = \mathbb{P}(W)$) and $\dim U = n-m$. Then $M \cong (\mathbb{C}^{n+1} \setminus W)/\mathbb{C}^* \cong \{ (w, u) \in W \oplus U \mid u \neq 0 \}/\mathbb{C}^*$.
  Define a deformation retraction from $M$ to the subspace $\{ (0, u) \in W \oplus U \mid u \neq 0 \}/\mathbb{C}^*$ by
  \[
  h_t([z_0:\cdots:z_n]) = [(1-t)z_0 : \cdots : (1-t)z_m : z_{m+1} : \cdots : z_n], \quad t \in [0,1].
  \]
  Therefore $M \simeq \mathbb{CP}^{n-m-1}$ and $H^*(\mathbb{CP}^{n-m-1})\cong\mathbb{Z}[x] / (x^{n-m})$, where $\deg x = 2$.