----- > [!proposition] Proposition. ([[(co)homology of the spheres]]) > [[singular homology|We]] [[singular cohomology|have]] $H_{i}(\mathbb{S}^{n})=H^{i}(\mathbb{S}^{n})= \begin{cases} \mathbb{Z} & i = 0, n \\ 0 & \text{else.} \end{cases}$ ^proposition > [!proposition] Corollary. > If $n \neq m$, then $\mathbb{S}^{n-1} \not \simeq \mathbb{S}^{m-1}$ (not [[homotopy equivalent]]), since the [[singular homology|homology groups]] differ. > If $n \neq m$, then $\mathbb{R}^{n} \not \cong \mathbb{R}^{m}$ (not [[homeomorphism|homeomorphic]]), for if so $\underbrace{ \mathbb{R}^{n}-\{ 0 \} }_{ \simeq \mathbb{S}^{n-1} } \cong \underbrace{ \mathbb{R}^{m}-\{ 0\} }_{ \simeq \mathbb{S}^{m-1} }$. ^proposition > [!proof]- Proof. ([[(co)homology of the spheres]]) > We know $H_{0}=\mathbb{Z}$ because [[connected]]. As for the rest, use [[Mayer-Vietoris theorem|Mayer-Vietoris]] + induction. Cover $\mathbb{S}^{1}$ like so: ![[Pasted image 20250610184737.png]] > > Applying MV, get a long [[exact sequence]] $\begin{align} > \cdots \to &\overbrace{ H_{1}(A \cap B) }^{ =0 } \to\overbrace{ H_{1}(A) \oplus H_{1}(B) }^{ =0 } \to H_{1}(\mathbb{S}^{1}) \\ > \xrightarrow{\partial_{MV}} &\underbrace{ H_{0}(A \cap B) }_{ \mathbb{Z}\langle p, q \rangle }\xrightarrow{\iota_{A_{*}}\oplus \iota_{B_{*}}} \underbrace{ H_{0}(A) }_{ \mathbb{Z} } \oplus \underbrace{ H_{0}(B) }_{ \mathbb{Z} } \to H_{0}(\mathbb{S}^{1}) > \end{align}$ > by exactness, $\partial_{MV}$ is [[injection|injective]], hence $H_{1}(\mathbb{S}^{1})$ is isomorphic to its image in $H_{0}(A \cap B) \cong \mathbb{Z}\langle p,q \rangle$, which equals the kernel of $\iota_{A_{*} }\oplus \iota_{B_{*}}$. Since the inclusion map sends each of $p$ and $q$ to the unique connected components comprising $A$ and $B$, matrix is $\begin{bmatrix}1 & 1 \\1 & 1\end{bmatrix}$ so kernel is generated by $p-q$. So $H_{1}(\mathbb{S}^{1}) \cong \mathbb{Z}\langle p-q \rangle$. > > For $\mathbb{S}^{n}$, cut up as $\begin{align} > A&=\mathbb{S}^{n}-\{ N \} \cong \mathbb{R}^{n} \simeq * \\ > B&= \mathbb{S}^{n} - \{ S \} \cong \mathbb{R}^{n} \simeq * > \end{align}$ > with $A \cap B \cong \mathbb{R} \times \mathbb{S}^{n-1} \simeq \mathbb{S}^{n-1}$. So we can 'induct up' using Mayer-Vietoris: $\begin{align} > \cdots \to & H_{i}(\mathbb{S}^{n-1}) \to H_{i}(*) \oplus H_{i}(*) \to H_{i}(\mathbb{S}^{n}) \\ > \xrightarrow{\partial_{MV}} &H_{i-1}(\mathbb{S}^{n-1}) \to H_{i-1}(*) \oplus H_{i-1}(*) \to H_{i-1}(\mathbb{S}^{n}) > \end{align}$ > Suppose $n \geq 2$ (we already did $n=1$). If $i>1$, then $H_{i}(*)=0=H_{i-1}(*)$ and hence $\partial_{MV}:H_{i}(\mathbb{S}^{n}) \to H_{i-1}(\mathbb{S}^{n-1})$ is an isomorphism. In particular, $H_{n}(\mathbb{S}^{n}) \cong H_{n-1}(\mathbb{S}^{n-1}) \cong \dots \cong H\underbrace{ _{1}(\mathbb{S}^{1}) }_{ \mathbb{Z} }$ > while for $1<i<n$ gets down to $H_{1}(\mathbb{S}^{n-i+1})$ > > All that remains is to look at $i=0,1$. The $i=0$ case is done because connected. For $i=1$, exercise. > > The computation for cohomology is similar (or immediate if willing to grant [[universal coefficients theorem for homology|UCT]]). ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```