----- > [!proposition] Proposition. ([[the degree of a reflection is -1]]) > A [[reflection]] $r:\mathbb{S}^{n} \to \mathbb{S}^{n}$ about a [[hyperplane]] $H$ has [[degree of a continuous map on the sphere|degree]] $-1$. ^proposition > [!proposition] Corollary. > [[degree of the antipodal map|The]] [[antipodal map]], as a composition of $n$ reflections, has degree $(-1)^{n}$. ^proposition > [!proof]- Proof. ([[the degree of a reflection is -1]]) > > [[cover|Cover]] $\mathbb{S}^{n}$ with $\begin{align} > A=\mathbb{S}^{n}-\{ N \} \cong \mathbb{R}^{n} \simeq * \\ > B = \mathbb{S}^{n} - \{ S \} \cong \mathbb{R}^{n} \simeq * > \end{align}$ > where both north and south poles are supposed to lie in the plane of reflection $H$. > > ![[Pasted image 20250510223659.png|200]] > > Then $r(A) = A$ and $r(B) = B$ as sets. $A \cap B$ is the doubly-punctured sphere, which can be [[homotopy equivalent|contracted]] to the copy of $\mathbb{S}^{n-1}$ that is the equator. We get a diagram > > ![[Pasted image 20250512173258.png]] > > which implies $\text{deg }r_{*}=\text{deg }r_{*} |_{\text{equator}}$. So by induction it suffices to do the case $n=1$. > > Here $A \cap B$ looks like two copies $p$ and $q$ of $\mathbb{R}$ (DC's image): > > ![[Pasted image 20250512174252.png|200]] > Now inspecting more with [[Mayer-Vietoris theorem|Mayer-Vietoris]], have (note $\partial_{MV}$ is an embedding) > ![[Pasted image 20250512175101.png]] > > Note (or [[(co)homology of the spheres|recall]]) $\iota_{A} \oplus \iota_{B}$ has kernel, which equals $H_{0}(A \cap B)$, generated by $p-q$. The middle vertical map swaps $p$ and $q$, sending $p \mapsto q$ and $q \mapsto p$. The blue is how we get the final result: > > ![[Pasted image 20250512175555.png]] > ----- #### ---- #### 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 > ```