Subsection 7.1.5. (0PH2) · ScalingStacks

7.1.5. Tangential multiplicity

Let $X$ be a $1$ -dimensional space. Let $x\in X$ and $U$ be a small open neighborhood of $x$ .

Let $c\in C(x)$ and let $U_{c}$ be the connected component of $U-\{x\}$ corresponding to $c$ . Given $\gamma$ a path in $X$ , let $I_{c}^{+}(\gamma)$ (resp. $I_{c}^{-}(\gamma)$ ) be the set of elements $t\in[0,1]$ such that $\gamma(t)=x$ and there is $\varepsilon>0$ with $t+\varepsilon<1$ and $\gamma((t,t+\varepsilon))\subset U_{c}$ (resp. $t-\varepsilon>0$ and $\gamma((t-\varepsilon,t))\subset U_{c}$ ).

When $\gamma$ is minimal, the set $I_{c}^{\pm}(\gamma)$ is finite and it follows from Lemma 7.1.18 that its cardinality depends only on the homotopy class $[\gamma]$ . We put $m_{c}^{\pm}([\gamma])=|I_{c}^{\pm}(\gamma)|\in{\mathbf{Z}}_{\geq 0}$ for $\gamma$ minimal and $m_{c}([\gamma])=m_{c}^{+}([\gamma])-m_{c}^{-}([\gamma])$ . Similarly, whether or not $0\in I_{c}^{+}$ depends only on the homotopy class $[\gamma]$ (for $\gamma$ minimal).

0P8X

Lemma 7.1.21. Let $\gamma$ be a path in $X$ such that $\gamma^{-1}(x)$ has finitely many connected components, none of which contain $0$ or $1$ in the closure of their interior.

We have $\partial(\gamma^{-1}(x))=\bigcup_{c\in C(x)}(I_{c}^{+}(\gamma)\cup I_{c}^{-}(\gamma))$ and $|I_{c}^{+}(\gamma)|-|I_{c}^{-}(\gamma)|=m_{c}([\gamma])$ for all $c\in C(x)$ .

0P8Y

Proof. The first statement is clear. Let us now prove the second statement. That statement is clear if $\gamma((0,1))\cap(X_{exc}\cup\{x\})=\emptyset$ .

The left side of the equality is additive under compositions of paths, and so is the right side by Lemma 7.1.22 below.

Assume now $\gamma^{-1}(X_{exc}\cup\{x\})$ is finite. The path $\gamma$ is a (finite) composition of paths mapping $(0,1)$ into the complement of $X_{exc}\cup\{x\}$ , hence the statement holds for $\gamma$ .

Consider now the general case. The proof of Lemma 7.1.16 for $E=X_{exc}\cup\{x\}$ produces a path $\gamma^{\prime}$ homotopic to $\gamma$ such that $\gamma^{\prime-1}(E)$ is finite and such that $|I_{c}^{+}(\gamma)|-|I_{c}^{-}(\gamma)|=|I_{c}^{+}(\gamma^{\prime})|-|I_{c}^{-}(\gamma^{\prime})|$ . Since the statement holds for $\gamma^{\prime}$ , it follows that it holds for $\gamma$ . ∎

Let $\zeta$ be the homotopy class of a minimal path $\gamma$ . Let $x=\zeta(0)$ . There is a unique $c\in C(x)$ such that $0\in I_{c}(\gamma)^{+}$ and we define $\zeta(0+)=\{c\}$ . Similarly, we define $\zeta(1-)=\{c^{\prime}\}$ , where $c^{\prime}\in C(\zeta(1))$ is unique such that $1\in I_{c}(\gamma)^{-}$ .

When $\zeta$ is the homotopy class of a constant path we put $\zeta(0+)=C(\zeta(0))$ , $\zeta(1-)=C(\zeta(1))$ and $m_{c}^{\pm}(\zeta)=m_{c}(\zeta)=0$ .

Given a category ${\mathcal{C}}$ , we denote by $H_{0}({\mathcal{C}})$ the abelian group generated by maps in ${\mathcal{C}}$ modulo the relation $f+g=f\circ g$ for any two composable maps $f$ and $g$ . We denote by $\llbracket f\rrbracket$ the class in $H_{0}({\mathcal{C}})$ of a map $f$ of ${\mathcal{C}}$ . Note that if $f$ is an identity map, then $\llbracket f\rrbracket=0$ .

Note that $H_{0}$ is left adjoint to the functor sending an abelian group to the category with one object with endomorphism monoid that abelian group.

Let $R(X)=H_{0}(\Pi(X))$ . Note that $R(X)$ is generated by the set $I$ of homotopy classes of paths $\gamma$ such that $\gamma$ is injective. It follows from the description of the composition of two minimal paths in §7.1.4 that $R(X)$ has a presentation with generating set the non-identity homotopy classes of paths and relations $[\gamma\circ\gamma^{\prime}]=[\gamma]+[\gamma^{\prime}]$ if $\gamma$ , $\gamma^{\prime}$ and $\gamma\circ\gamma^{\prime}$ are minimal and $[\gamma]+[\gamma^{-1}]=0$ for $\gamma$ minimal. Note finally that every element of $R(X)$ is a linear combination of non-identity homotopy classes of paths such that the intersection between the supports of two distinct homotopy classes is finite.

0P8Z

Lemma 7.1.22. Given $c\in T(X)$ , the map $m_{c}$ induces a morphism of groups $R(X)\to{\mathbf{Z}}$ .

0P90

Proof. Consider $\gamma$ and $\gamma^{\prime}$ two injective composable paths such that $\gamma\circ\gamma^{\prime}$ is injective. We have $m_{c}^{\pm}([\gamma\gamma^{\prime}])=m_{c}^{\pm}([\gamma])+m_{c}^{\pm}([\gamma^{\prime}])$ .

Consider now $\gamma$ a minimal path. We have $m_{c}^{\pm}([\gamma])=m_{c}^{\mp}([\gamma^{-1}])$ , hence $m_{c}([\gamma])+m_{c}([\gamma^{-1}])=0=m_{c}([\gamma^{-1}\circ\gamma])$ . The lemma follows. ∎

The next lemma shows how to realize $R(X)$ as a subgroup of the group of maps $U\to{\mathbf{Z}}$ , where $U$ is a dense subset of $X$ .

0P91

Lemma 7.1.23. Let $U$ be a dense subset of $X-(\partial X\cup X_{exc})$ . Given $x\in U$ , fix a group morphism $l_{x}:{\mathbf{Z}}^{C(x)}\to{\mathbf{Z}}$ that does not factor through the sum map.

The morphism $(l_{x}\circ(m_{c})_{c\in C(x)})_{x\in U}:R(X)\to{\mathbf{Z}}^{U}$ is injective.

0P92

Proof. Let $L$ be a non-empty finite subset of $I$ such that $\operatorname{supp}\nolimits(\zeta)\cap\operatorname{supp}\nolimits(\zeta^{\prime})$ is finite for any two distinct elements $\zeta$ and $\zeta^{\prime}$ in $L$ . Let $r=\sum_{\zeta\in L}a_{\zeta}\llbracket\zeta\rrbracket$ where $a_{\zeta}\in{\mathbf{Z}}-\{0\}$ for $\zeta\in L$ . Let $\zeta_{0}\in L$ . There is $x\in\operatorname{supp}\nolimits(\zeta_{0})\cap U$ with $x{\not\in}\{\zeta_{0}(0),\zeta_{0}(1)\}$ and $x{\not\in}\bigcup_{\zeta\in L-\{\zeta_{0}\}}\operatorname{supp}\nolimits(\zeta)$ . Let $c\in C(x)$ and $\iota(c)$ be the other element of $C(x)$ . We have $m_{c}(\zeta_{0})=-m_{\iota(c)}(\zeta_{0})=\pm 1$ , while $m_{c}(\zeta^{\prime})=m_{\iota(c)}(\zeta^{\prime})=0$ for $\zeta^{\prime}\in L-\{\zeta_{0}\}$ . It follows that $m_{c}(r)=-m_{\iota(c)}(r)=\pm a_{\gamma}$ . Consequently, $\bigl(l_{x}\circ(m_{c},m_{\iota(c)})\bigr)(r)=\pm l_{x}(a_{\gamma},-a_{\gamma})\neq 0$ . Since every non-zero element of $R(X)$ is of the form $r$ as above, the lemma follows. ∎

Let $f:X\to X^{\prime}$ be a morphism of $1$ -dimensional spaces. The next lemma follows from the injectivity statement of Lemma 7.1.8.

0P93

Lemma 7.1.24. Given $x\in X$ , $c\in C(X)$ and $\zeta$ a homotopy class of paths in $X$ , we have $m_{f(c)}^{\pm}(f(\zeta))=m_{c}^{\pm}(\zeta)$ and $m_{f(c)}(f(\zeta))=m_{c}(\zeta)$ .

Note that $f$ induces a morphism of groups $f:R(X)\to R(X^{\prime})$ .

0P94

Lemma 7.1.25. Let $H$ be the subgroup of $R(X^{\prime})$ generated by classes $[\gamma]$ with $\operatorname{supp}\nolimits(\gamma)\subset\overline{X^{\prime}-f(X)}$ .

The composition $R(X)\xrightarrow{f}R(X^{\prime})\xrightarrow{{\mathrm{can}}}R(X^{\prime})/H$ is injective.

0P95

Proof. Let $U^{\prime}=X^{\prime}-(X^{\prime}_{f}\cup X^{\prime}_{exc}\cup\partial X^{\prime})$ , a dense subset of $X^{\prime}$ . Note that $U=f^{-1}(U^{\prime})$ is a dense subset of $X-(X_{exc}\cup\partial X)$ . Given $x^{\prime}\in U^{\prime}$ , fix a morphism $l_{x^{\prime}}:{\mathbf{Z}}^{C(x^{\prime})}\to{\mathbf{Z}}$ that does not factor through the sum map. Given $x\in U$ , let $l_{x}=l_{x^{\prime}}\circ f:{\mathbf{Z}}^{C(x)}\to{\mathbf{Z}}$ . Lemma 7.1.23 shows that $(l_{x}\circ(m_{c})_{c\in C(x)})_{x\in U}:R(X)\to{\mathbf{Z}}^{U}$ is injective. This map is equal to the composition

R(X)\xrightarrow{f}R(X^{\prime})\xrightarrow{(l_{x^{\prime}}\circ(m_{c^{\prime}})_{c^{\prime}\in C(x^{\prime})})_{x^{\prime}\in U^{\prime}}}{\mathbf{Z}}^{U^{\prime}}\xrightarrow{f^{*}}{\mathbf{Z}}^{U}

since $m_{f(c)}^{\pm}(f(\zeta))=m_{c}^{\pm}(\zeta)$ and $m_{f(c)}(f(\zeta))=m_{c}(\zeta)$ for all $x\in X$ , $c\in C(X)$ and all homotopy classes of paths $\zeta$ in $X$ (Lemma 7.1.24). Since $H$ is contained in the kernel of the composition

R(X^{\prime})\xrightarrow{(l_{x^{\prime}}\circ(m_{c^{\prime}})_{c^{\prime}\in C(x^{\prime})})_{x^{\prime}\in U^{\prime}}}{\mathbf{Z}}^{U^{\prime}}\xrightarrow{f^{*}}{\mathbf{Z}}^{U},

it follows that the composite map of the lemma is injective. ∎

Given $M$ a subset of $X$ , we denote by $R_{M}(X)$ the subgroup of $R(X)$ generated by classes of paths $\gamma$ with endpoints in $M$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2