---- > [!theorem] Theorem. ([[the nulhomotopy characterization on the circle]]) > Let $h: \mathbb{S}^{1} \to X$ be a [[continuous]] map from the [[sphere|circle]] (i.e., a [[parameterized curve|loop]]). Then the follow conditions are equivalent: >1. $h$ is [[homotopy|nulhomotopic]]; >2. $h$ extends to a [[continuous]] map $k:B^{2} \to X$ on the whole disc; >3. $h_{*}$ is the [[group homomorphism|trivial homomorphism]] of [[fundamental group|fundamental groups]]. > [!proof]- Proof. ([[the nulhomotopy characterization on the circle]]) > $(1) \implies (2)$. Suppose $h$ is [[homotopy|nulhomotopic]], let $H:\mathbb{S}^{1} \times I \to X$ be a [[homotopy]] between $h$ and a constant map. Let $\pi:\mathbb{S}^{1} \times I \to B^{2}$ be the map $\pi(x,t)=(1-t)x.$ > $\pi$ is [[continuous]], [[open map|closed]], and [[surjection|surjective]], so it is a [[topological quotient map|quotient map]]; it collapses $\mathbb{S}^{1} \times \{ 1 \}$ to the point $\b 0$ and is otherwise [[injection|injective]] (you can imagine it as 'filling in' the circle with concentric circles of decreasing radius). So in order to employ the [[universal property of quotient topology]], all we must do is check that that $H$ is constant on $\mathbb{S}^{1} \times \{ 1 \}= \pi ^{-1}(\boldsymbol 0)$, which it must be because $H$ is nulhomotopic. Thus, a [[continuous]] map $k$ is induced from $\mathbb{B}^{2}$ to $X$ which agrees with $h$ on $\mathbb{S}^{1}$. See picture in Munkres, page 349. > > $(2) \implies (3)$. With $\iota:\mathbb{S}^{1} \to \mathbb{B}^{2}$ denoting the [[inclusion map]], $h$ equals the composite $k \circ \iota$. By functoriality, $h_{*}=k_{*} \circ \iota_{*}$. Since the [[fundamental group]] of $\mathbb{B}^{2}$ is trivial, $k_{*}$ is trivial. > > > $(3) \implies (1)$. We need to obtain a [[homotopy]] between $h$ and a constant map. Take a loop $[p_{0}]$ generating $\pi_{1}(\mathbb{S}^{1}, b_{0})$. The loop $h \circ p_{0}$ represents the identity element of $\pi_{1}(X, x_{0})$, where $x_{0}=h(b_{0})$ (since $h_{*}$ is trivial). So there is a [[path homotopy]] $F$ in $X$ between $h \circ p_{0}$ and the constant map $x_{0}$. The map $p_{0} \times \id: I \times I \to \mathbb{S}^{1} \times I$ is a [[topological quotient map]] ([[continuous]], closed, surjective). It is injective except for mapping $(0,t)$ and $(1,t)$ to $(b_{0},t)$ for each $t$. $F$ is constant on this data, since it is a homotopy. The universal property then induces a map $H$ which is the desired homotopy. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ``` #reformatrevisebatch04