---- > [!theorem] Theorem. ([[Brouwer's fixed point theorem]]) > Any [[continuous|map]] $f:\mathbb{D}^{n} \to \mathbb{D}^{n}$ has a fixed point. ^theorem > [!proof]- Proof. ([[Brouwer's fixed point theorem]]) > Suppose not. Define $r:\mathbb{D}^{n} \to \mathbb{S}^{n-1}=\partial \mathbb{D}^{n}$ by taking the intersection of the [[ray]] from $f(x)$ through $x$ with $\partial \mathbb{D}^{n}$. This is [[continuous]], as was justified in Math 590 Homework (and somewhere in these notes, but in a hurry right now, sorry). > > ![[Pasted image 20250508225204.png]] > > If $x \in \partial \mathbb{D}^{n}$, then $f(x)=x$. So we have a restricted map $\mathbb{S}^{n-1} = \partial \mathbb{D}^{n} \xhookrightarrow{\iota} \mathbb{D}^{n} \xrightarrow{r} \partial \mathbb{D}^{n}=\mathbb{S}^{n-1}$ > which is just the [[identity map|identity]] on $\mathbb{S}^{n-1}$. That's a contradiction: [[singular (co)chain map and homomorphism induced by a continuous map|applying]] $H_{n-1}(-)$ yields $H_{n-1}(\mathbb{S}^{n-1}) \xrightarrow{\iota_{*}} \cancel{ H_{n-1}(\mathbb{D}^{n}) }^{=0} \xrightarrow{r_{*}} H_{n-1}(\mathbb{S}^{n-1})$ > and $\id_{\mathbb{S}^{n-1}}$ should induce the identity map on homology, not the zero map. > ---- #### ----- #### 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 > ```