2.1 Definition of the Hurwitz–Moore spaces

We first introduce notation for rectangles and horizontal strips in the plane.

Notation 2.3 For $t\geq 0$ we denote by ${\mathcal {R}}_t\subset \mathbb {H}$ the standard closed rectangle $[0,t]\times [0,1]$ of width $t$ and height 1; we also denote ${\mathcal {R}}_{\infty }=[0,\infty )\times [0,1]$ the half-infinite, closed strip and by ${\mathcal {R}}_{\mathbb {R}}=(-\infty,+\infty )\times [0,1]$ the infinite, closed strip.

For $0\leq t\leq +\infty$ we denote by $\mathring {{\mathcal {R}}}_t=(0,t)\times (0,1)$ the standard open rectangle (or half-infinite strip) of width $t$ and height 1 and by $\mathring {{\mathcal {R}}}_{\mathbb {R}}=(-\infty,+\infty )\times (0,1)$ the infinite open strip. Also let $\partial {\mathcal {R}}_t={\mathcal {R}}_t\setminus \mathring {{\mathcal {R}}}_t$ for all $0\leq t\leq \infty$ and $\partial {\mathcal {R}}_{\mathbb {R}}={\mathcal {R}}_{\mathbb {R}}\setminus \mathring {{\mathcal {R}}}_{\mathbb {R}}$, denote the boundary of ${\mathcal {R}}_t$ and ${\mathcal {R}}_{\mathbb {R}}$, respectively. We use the abbreviations $({\mathcal {R}}_t,\partial )$ and $({\mathcal {R}}_{\mathbb {R}},\partial )$ for the nice couples $({\mathcal {R}}_t,\partial {\mathcal {R}}_t)$ and $({\mathcal {R}}_{\mathbb {R}},\partial {\mathcal {R}}_{\mathbb {R}})$, respectively.

For $0\leq t\leq +\infty$ we denote by $\breve {{\mathcal {R}}}_t=(0,t)\times [0,1]$ the standard, vertically closed rectangle (or vertically closed half-infinite strip) of width $t$ and height 1 and by $\breve {\partial }\breve {{\mathcal {R}}}_t=(0,t)\times \{0,1\}\subset \breve {{\mathcal {R}}}_t$ the horizontal boundary of $\breve {{\mathcal {R}}}$. Similarly, we denote $\breve {{\mathcal {R}}}_{\mathbb {R}}={\mathcal {R}}_{\mathbb {R}}$ and $\breve {\partial }\breve {{\mathcal {R}}}_{\mathbb {R}}=\partial \breve {{\mathcal {R}}}_{\mathbb {R}}=(-\infty,+\infty )\times \{0,1\}$. We use the abbreviations $(\breve {{\mathcal {R}}}_t,\breve {\partial })$ and $(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })$ for the nice couples $(\breve {{\mathcal {R}}}_t,\breve {\partial }\breve {{\mathcal {R}}}_t)$ and $(\breve {{\mathcal {R}}}_{\mathbb {R}};\breve {\partial }\breve {{\mathcal {R}}}_{\mathbb {R}})$, respectively.

Whenever $t=1$ we drop it from the notation, so we abbreviate ${\mathcal {R}}_1$ as ${\mathcal {R}}$, $\mathring {{\mathcal {R}}}_1$ as $\mathring {{\mathcal {R}}}$ and $\breve {{\mathcal {R}}}_1$ as $\breve {{\mathcal {R}}}$. See Figure 1.

Figure 1. Left: the rectangle $\mathring {{\mathcal {R}}}_{1/2}$. Right: the nice couple $(\breve {{\mathcal {R}}}_{3/4},\breve {\partial }\breve {{\mathcal {R}}}_{3/4})$.

Note that $\breve {{\mathcal {R}}}_0=\mathring {{\mathcal {R}}}_0=\emptyset$. For $t\leq t'$ the identity of $\mathbb {C}$ restricts to an inclusion $\mathring {{\mathcal {R}}}_t\subset \mathring {{\mathcal {R}}}_{t'}$ and induces an inclusion of Hurwitz spaces $\mathrm {Hur}(\mathring {{\mathcal {R}}}_t)\subseteq \mathrm {Hur}(\mathring {{\mathcal {R}}}_{t'})$.

Note that, for fixed $t\geq 0$, the slice of $\mathring {\mathrm {HM}}$ containing couples of the form $(t,\mathfrak {c})$ is homeomorphic to $\mathrm {Hur}(\mathring {{\mathcal {R}}}_t)$: thus, $\mathring {\mathrm {HM}}$, as a set, is the disjoint union $\bigsqcup _{t\geq 0}\mathrm {Hur}(\mathring {{\mathcal {R}}}_t)$. Similarly, $\breve {\mathrm {HM}}$ is in natural bijection with the set $\bigsqcup _{t\geq 0}\mathrm {Hur}(\breve {{\mathcal {R}}}_t,\breve {\partial })$.

Note that $(\emptyset,\mathbb {1},\mathbb {1})$ is the only point in the spaces $\mathrm {Hur}(\mathring {{\mathcal {R}}}_0)$ and $\mathrm {Hur}(\breve {{\mathcal {R}}}_0,\breve {\partial })$, since ${\mathring {{\mathcal {R}}}_0=\breve {{\mathcal {R}}}_0=\emptyset}$. In other words, we have $\mathrm {Hur}_+(\mathring {{\mathcal {R}}}_0)=\mathrm {Hur}_+(\breve {{\mathcal {R}}}_0,\breve {\partial })=\emptyset$.

Notation 2.6 We write $\breve {\mathrm {HM}}$ as the disjoint union $[0,\infty )\times \{(\emptyset,\mathbb {1},\mathbb {1})\}\sqcup \breve {\mathrm {HM}}_+$, where we set

\[ \breve{\mathrm{HM}}_+\colon=([0,\infty)\times\mathrm{Hur}_+(\breve{{\mathcal{R}}}_{\infty},\breve{\partial}))\cap\breve{\mathrm{HM}} \subset [0,\infty)\times\mathrm{Hur}(\breve{{\mathcal{R}}}_{\infty},\breve{\partial}). \]

By the previous discussion, every couple $(t,\mathfrak {c})\in \breve {\mathrm {HM}}_+$ satisfies $t>0$.

Proof. The proof is almost identical in the two cases, so we will focus on the second case, which is slightly more difficult.

For $s>0$ the map $\Lambda _s\colon \mathbb {C}\to \mathbb {C}$ given by $\Lambda _s(z)=(s\Re (z),\Im (z))$ is a morphism of nice couples $\Lambda _s\colon (\breve {{\mathcal {R}}}_\infty,\breve {\partial })\to (\breve {{\mathcal {R}}}_{\infty },\breve {\partial })$. Putting all values of $s\geq 0$ together we obtain a continuous map $\Lambda \colon \mathbb {C}\times (0,\infty )\to \mathbb {C}$; by [Reference BianchiBia23a, Proposition 4.4] we obtain a continuous map

\[ \Lambda_*\colon \mathrm{Hur}(\breve{{\mathcal{R}}}_\infty,\breve{\partial})\times (0,\infty)\to \mathrm{Hur}(\breve{{\mathcal{R}}}_\infty,\breve{\partial}). \]

Define $\tilde \Lambda \colon (0,\infty )\times \mathrm {Hur}(\breve {{\mathcal {R}}}_{\infty },\breve {\partial })\times (0,\infty )\to (0,\infty )\times \mathrm {Hur}(\breve {{\mathcal {R}}}_{\infty },\breve {\partial })$ by the formula $\tilde \Lambda (t,\mathfrak {c};s)=(ts,\Lambda _*(\mathfrak {c},s))$. We are now able to define a homotopy ${\mathcal {H}}^{\Lambda }\colon \breve {\mathrm {HM}}\times [0,1]\to \breve {\mathrm {HM}}$ by setting

\[ {\mathcal{H}}^{\Lambda}(t,\mathfrak{c};s)=\left\{\begin{array}{@{}ll} (ts+1-s,\tilde\Lambda(\mathfrak{c},(ts+1-s)/t)) & \text{for all }t>0\text{ and }\mathfrak{c}\in\breve{\mathrm{HM}}_+;\\ (ts+1-s,(\emptyset,\mathbb {1},\mathbb {1})) & \text{for }\mathfrak{c}=(\emptyset,\mathbb {1},\mathbb {1}) \text{ and all }t\geq 0. \end{array}\right. \]

Note that ${\mathcal {H}}^{\Lambda }((1,\mathfrak {c}),s)=(1,\mathfrak {c})$ for all $\mathfrak {c}\in \mathrm {Hur}(\breve {{\mathcal {R}}},\breve {\partial })$, including $(\emptyset,\mathbb {1},\mathbb {1})$ and all $0\leq s\leq 1$; moreover, the map ${\mathcal {H}}(-;1)$ is the identity of $\breve {\mathrm {HM}}$, whereas the map ${\mathcal {H}}^{\Lambda }(-;0)$ has image inside $\mathrm {Hur}(\breve {{\mathcal {R}}},\breve {\partial })$.

The reason for the name Moore in Definition 2.4 is that, as we will see in § 2.2, there is a natural structure of topological monoid on both $\mathring {\mathrm {HM}}$ and $\breve {\mathrm {HM}}$. In contrast, $\mathrm {Hur}(\mathring {{\mathcal {R}}})$ and $\mathrm {Hur}(\breve {{\mathcal {R}}})$ are only endowed with the structure of $E_1$-algebras in a natural way. The spaces $\mathring {\mathrm {HM}}$ and $\breve {\mathrm {HM}}$ play the role of the strictification of the $E_1$-algebras $\mathrm {Hur}(\mathring {{\mathcal {R}}})$ and $\mathrm {Hur}(\breve {{\mathcal {R}}})$ to actual topological monoids, just as the Moore loop space $\Omega ^{\mathrm {Moore}}X$ of a pointed topological space $X$ is a strictly associative and strictly unital replacement of the $E_1$-algebra given by the usual loop space $\Omega X$.

2.2 Topological monoid structure

In this subsection we define a topological monoid structure on $\mathring {\mathrm {HM}}$ and $\breve {\mathrm {HM}}$. For $\mathring {\mathrm {HM}}$ the basic idea is to juxtapose two configurations $\mathfrak {c}\in \mathrm {Hur}(\mathring {{\mathcal {R}}}_t)$ and $\mathfrak {c}'\in \mathrm {Hur}(\mathring {{\mathcal {R}}}_{t'})$ to obtain a larger configuration supported on the rectangle $\mathring {{\mathcal {R}}}_{t+t'}$; for $\breve {\mathrm {HM}}$ the idea is similar, but using vertically closed rectangles and juxtaposing also their horizontal boundaries.

In the entire subsection we focus on $\mathring {\mathrm {HM}}$ and write in parentheses the changes needed in the analogous discussion about $\breve {\mathrm {HM}}$. Whenever we write $\mathfrak {Q}(P)$ for a subset $P\subset \mathbb {H}$, we use Notation 2.1 with $\mathbb {Y}=\emptyset$ (respectively, $\mathbb {Y}=\breve {\partial }\breve {{\mathcal {R}}}_{\mathbb {R}}$, see Notation 2.3).

Note that for $P,t,t'$ as in Notation 2.8, if $P\subset \mathring {{\mathcal {R}}}_{t'}$ (respectively $P\subset \breve {{\mathcal {R}}}_{t'})$, then $P+t\subset \mathring {{\mathcal {R}}}_{t'}+t\subset \mathring {{\mathcal {R}}}_{t+t'}$ (respectively, $P+t\subset \breve {{\mathcal {R}}}_{t'}+t\subset \breve {{\mathcal {R}}}_{t+t'}$).

Definition 2.9 (Definition 6.7 in [Reference BianchiBia23a])

For $t\in \mathbb {R}$ we define a homeomorphism $\tau _t\colon (\mathbb {C},*)\to (\mathbb {C},*)$ by

\[ \tau_t(z)=\left\{\begin{array}{@{}ll} z & \text{if }\Im(z)\leq -1,\\ z+t & \text{if } \Im(z)\geq 0,\\ z+(\Im(z)+1)t & \text{if }{-}1\leq \Im(z)\leq 0. \end{array}\right. \]

Note that for $t,t'\geq 0$ we have $\tau _t(\mathring {{\mathcal {R}}}_{t'})=\mathring {{\mathcal {R}}}_{t'}+t$ (respectively, $\tau _t(\breve {{\mathcal {R}}}_{t'},\breve {\partial })=(\breve {{\mathcal {R}}}_{t'},\breve {\partial })+t$).

Notation 2.10 (Notation 6.8 in [Reference BianchiBia23a])

For $t\in \mathbb {R}$ we denote by $\mathbb {C}_{\Re \geq t}\subset \mathbb {C}$ the subspace containing all $z\in \mathbb {C}$ with $\Re (z)\geq t$. Similarly, we define $\mathbb {C}_{\Re >t}$, $\mathbb {C}_{\Re \leq t}$, $\mathbb {C}_{\Re < t}$ and $\mathbb {C}_{\Re =t}$, the latter being a vertical line. For all $-\infty \leq t\leq t'\leq +\infty$ we define a subspace $\mathbb {S}_{t,t'}\subset \mathbb {C}$ by

\[ \mathbb{S}_{t,t'}=\tau_t(\mathbb{C}_{\Re\geq 0})\cap\tau_{t'}(\mathbb{C}_{\Re\leq 0}), \]

where we use the conventions $\tau _{-\infty }(\mathbb {C}_{\Re \geq 0})=\tau _{+\infty }(\mathbb {C}_{\Re \leq 0})=\mathbb {C}$ and $\tau _{-\infty }(\mathbb {C}_{\Re \leq 0})=\tau _{+\infty }(\mathbb {C}_{\Re \geq 0})=\emptyset$.

For all $t,t'\geq 0$, we have that $\mathbb {S}_{0,t+t'}$ is contractible and can be written as the union of the contractible spaces $\mathbb {S}_{0,t}$ and $\mathbb {S}_{t,t+t'}$ along the contractible space $\mathbb {S}_{t,t}$. Moreover, $\mathring {{\mathcal {R}}}_t\subset \mathring {\mathbb {S}}_{0,t}$ (respectively, $\breve {{\mathcal {R}}}_t\subset \mathring {\mathbb {S}}_{0,t}$), whereas $\mathring {{\mathcal {R}}}_{t'}+t\subset \mathring {\mathbb {S}}_{t,t+t'}$ (respectively, $\breve {{\mathcal {R}}}_{t'}+t\subset \mathring {\mathbb {S}}_{t,t+t'}$). Note also that $\tau _t$ restricts to a homeomorphism $\mathbb {S}_{0,t'}\to \mathbb {S}_{t,t+t'}$.

Recall [Reference BianchiBia23a, Definitions 3.15 and 3.16]: if $\mathbb {T}\subseteq \mathbb {C}$ is a contractible subspace containing $*=-\sqrt {-1}$, then for any nice couple of subspaces ${\mathcal {Y}}\subseteq {\mathcal {X}}\subseteq \mathring {\mathbb {T}}$ we can give an alternative definition of $\mathrm {Hur}({\mathcal {X}},{\mathcal {Y}})$, denoted by $\mathrm {Hur}^{\mathbb {T}}({\mathcal {X}},{\mathcal {Y}})$, using $\mathbb {T}$ instead of the entire $\mathbb {C}$ as ‘ambient space’: indeed, for any finite set $P\subseteq {\mathcal {X}}$, the fundamental group $\pi _1(\mathbb {T}\setminus P,*)$ is canonically identified with $\mathfrak {G}(P)=\pi _1(\mathbb {C}\setminus P,*)$ and similarly for fundamental PMQs. We thus get an identification $\mathfrak {i}^{\mathbb {C}}_{\mathbb {T}}\colon \mathrm {Hur}({\mathcal {X}},{\mathcal {Y}})\overset {\cong }{\to }\mathrm {Hur}^{\mathbb {T}}({\mathcal {X}},{\mathcal {Y}})$.

If, moreover, $\xi \colon (\mathbb {C},*)\to (\mathbb {C},*)$ is a semialgebraic and orientation-preserving homeomorphism of the plane preserving the upper half-plane $\mathbb {H}$ and if $\mathbb {T}',{\mathcal {X}}',{\mathcal {Y}}'$ are three other subspaces of $\mathbb {C}$ as above such that $\xi$ maps $\mathbb {T}\to \mathbb {T}'$, ${\mathcal {X}}\to {\mathcal {X}}'$ and ${\mathcal {Y}}\to {\mathcal {Y}}'$, then $\xi$ induces a map $\xi _*\colon \mathrm {Hur}^{\mathbb {T}}({\mathcal {X}},{\mathcal {Y}})\to \mathrm {Hur}^{\mathbb {T}'}({\mathcal {X}},{\mathcal {Y}})$.

There is finally a ‘disjoint union’ map $-\sqcup -\colon \mathrm {Hur}^{\mathbb {T}_1}({\mathcal {X}}_1,{\mathcal {Y}}_1)\times \mathrm {Hur}^{\mathbb {T}_2}({\mathcal {X}}_2,{\mathcal {Y}}_2)\to \mathrm {Hur}^{\mathbb {T}_1\cup \mathbb {T}_2}({\mathcal {X}}_1\sqcup {\mathcal {X}}_2,{\mathcal {Y}}_1\sqcup {\mathcal {Y}}_2)$, defined when $\mathbb {T}_1\cap \mathbb {T}_2$ is contractible and disjoint from both ${\mathcal {X}}_1,{\mathcal {X}}_2$ (in particular, this implies that ${\mathcal {X}}_1$ and ${\mathcal {X}}_2$ are disjoint).

Definition 2.11 For $t,t'\geq 0$ we define the maps $\mu _{t,t'}\colon \mathrm {Hur}(\mathring {{\mathcal {R}}}_t)\times \mathrm {Hur}(\mathring {{\mathcal {R}}}_{t'})\to \mathrm {Hur}(\mathring {{\mathcal {R}}}_{t+t'})$ and $\mu _{t,t'}\colon \mathrm {Hur}(\breve {{\mathcal {R}}}_t,\breve {\partial })\times \mathrm {Hur}(\breve {{\mathcal {R}}}_{t'},\breve {\partial })\to \mathrm {Hur}(\breve {{\mathcal {R}}}_{t+t'},\breve {\partial })$ as the following compositions:

Definition 2.12 Recall Definition 2.11. We define a map of sets

\[ \mu\colon \mathring{\mathrm{HM}}\times\mathring{\mathrm{HM}}\to\mathring{\mathrm{HM}} \quad(\text{respectively, } \mu\colon \breve{\mathrm{HM}}\times\breve{\mathrm{HM}}\to\breve{\mathrm{HM}} ) \]

by the formula $\mu (\,(t,\mathfrak {c}),\,(t',\mathfrak {c}')\,)=(t+t',\,\mu _{t,t'}(\mathfrak {c},\mathfrak {c}'))$. See Figure 2.

Figure 2. Left: two configurations in $\mathrm {Hur}^{\mathbb {S}_{0,1/2}}(\breve {{\mathcal {R}}}_{1/2},\breve {\partial })\cong \mathrm {Hur}(\breve {{\mathcal {R}}}_{1/2},\breve {\partial })\subset \breve {\mathrm {HM}}$. Right: their product in $\mathrm {Hur}^{\mathbb {S}_{0,1}}(\breve {{\mathcal {R}}},\breve {\partial })\cong \mathrm {Hur}(\breve {{\mathcal {R}}},\breve {\partial })\subset \breve {\mathrm {HM}}$.

The proof of Proposition 2.13 is in Appendix A.1.

Recall the notion of total monodromy from [Reference BianchiBia23a, Definitions 6.1 and 6.3]: for a generic nice couple $({\mathcal {X}},{\mathcal {Y}})$ we have a map $\omega \colon \mathrm {Hur}({\mathcal {X}},{\mathcal {Y}})\to G$ sending a configuration $(P,\psi,\varphi )$ to the value of the monodromy $\psi$ at the ‘large loop’, i.e. the element of $\mathfrak {G}(P)$ represented by a simple loop spinning clockwise around all points of $P$.

If ${\mathcal {Y}}=\emptyset$, one can lift this to a total monodromy $\hat \omega \colon \mathrm {Hur}({\mathcal {X}})\to \hat {{\mathcal {Q}}}$, where $\hat {{\mathcal {Q}}}$ is the completion of the PMQ ${\mathcal {Q}}$, as in [Reference BianchiBia21, Definition 2.19]. Concretely, $\hat {{\mathcal {Q}}}$ can be defined as the free, non-unital monoid generated by elements $\hat a$ for $a\in {\mathcal {Q}}$, satisfying $\hat a\hat b=\hat b\widehat {a^b}$ for all $a,b\in {\mathcal {Q}}$ and satisfying $\hat a\hat b=\widehat {ab}$ for all $a,b\in {\mathcal {Q}}$ such that the product $ab$ is already defined in ${\mathcal {Q}}$. The non-unital monoid $\hat {{\mathcal {Q}}}$ happens to have a unit, namely $\hat {\mathbb {1}}$ and a natural binary operation of conjugation can be defined on it, so that it becomes a PMQ with complete product; there is a natural inclusion of PMQs ${\mathcal {Q}}\hookrightarrow \hat {{\mathcal {Q}}}$, which is the universal map from ${\mathcal {Q}}$ to a complete PMQ. In the lift $\hat \omega$ of $\omega$ we need $\hat {{\mathcal {Q}}}$ rather than ${\mathcal {Q}}$ as target because the large loop in ${\mathcal {G}}(P)$ is not, in general, an element of the fundamental PMQ $\mathfrak {Q}(P)$ (unless $P$ is a singleton), so we cannot directly evaluate $\psi$ on it; but we can factor the large loop as a product of elements in $\mathfrak {Q}(P)$, evaluate $\psi$ on the factors and compute in $\hat {{\mathcal {Q}}}$ the corresponding product of elements of ${\mathcal {Q}}$.

Notation 2.14 For $(t,\mathfrak {c})$ and $(t',\mathfrak {c}')$ in $\mathring {\mathrm {HM}}$ (in $\breve {\mathrm {HM}}$) we denote by $(t,\mathfrak {c})\cdot (t',\mathfrak {c}')$ the configuration $\mu ((t,\mathfrak {c}),(t',\mathfrak {c}'))$.

We denote by $\hat \omega \colon \mathring {\mathrm {HM}}\to \hat {{\mathcal {Q}}}$ (respectively, $\omega \colon \breve {\mathrm {HM}}\to G$) the composition

where the first map is the projection on the second component and $\hat {{\mathcal {Q}}}$ denotes the completion of the PMQ ${\mathcal {Q}}$.

2.3 Computation of $\pi _0(\mathring {\mathrm {HM}})$

In this subsection we study the discrete monoid of path components of $\mathring {\mathrm {HM}}$. We will prove the following theorem, which is similar to [Reference BianchiBia23a, Proposition 6.4].

For a space $X$ we denote by $\pi _0\colon X\to \pi _0(X)$ the map assigning to each point of $X$ its path component. We denote by $\cdot$ the product of the discrete monoid $\pi _0(\mathring {\mathrm {HM}})$.

The proof of Lemma 2.18 is in Appendix A.2.

Proof of Theorem 2.15 First we prove that $\hat \omega \colon \mathring {\mathrm {HM}}\to \hat {{\mathcal {Q}}}$ is a map of monoids. Let $(t,\mathfrak {c}),(t',\mathfrak {c}')\in \mathring {\mathrm {HM}}$ and use Notation 2.2: we can choose simple loops $\gamma \subset \mathbb {S}_{-\infty,t}$ and $\gamma '\subset \mathbb {S}_{t,+\infty }$, spinning clockwise around $P$ and $P'+t$, respectively; the product $[\gamma ]\cdot [\gamma ']\in \mathfrak {G}(P\cup (P'+t))$ is represented by a simple loop spinning clockwise around $P\cup (P'+t)$. Denoting $(t,\mathfrak {c})\cdot (t',\mathfrak {c}')=(t+t',(P'',\psi ''))$, by definition of $\psi ''$ we have

\[ \hat\omega((t,\mathfrak{c})\cdot(t',\mathfrak{c}'))= \psi''([\gamma]\cdot[\gamma'])=\psi([\gamma])\cdot\psi'([\gamma']) =\hat\omega(t,\mathfrak{c})\cdot\hat\omega(t'\mathfrak{c}')\in\hat{{\mathcal{Q}}}. \]

Note that $\hat \omega ((0,(\emptyset,\mathbb {1},\mathbb {1})))=\mathbb {1}$, so $\hat \omega$ is a map of unital monoids; moreover, $\hat \omega (1,\mathfrak {c}_a)=\hat a\in \hat {{\mathcal {Q}}}$ for all $a\in {\mathcal {Q}}$, so that $\hat \omega \colon \pi _0(\mathring {\mathrm {HM}})\to \hat {{\mathcal {Q}}}$ hits the generators of $\hat {{\mathcal {Q}}}$ and is thus surjective.

By Lemma 2.18, the corresponding relations among the elements $\pi _0(1,\mathfrak {c}_a)\in \pi _0(\mathring {\mathrm {HM}})$ hold, so that the assignment $\hat a\mapsto \pi _0(1,\mathfrak {c}_a)$ defines a map of non-unital monoids $\Omega \colon \hat {{\mathcal {Q}}}\to \pi _0(\mathring {\mathrm {HM}})$; note that, though both the source and the target of $\Omega$ are indeed unital monoids, $\pi _0(1,\mathfrak {c}_\mathbb {1})=\Omega (\mathbb {1})$ is not the unit of $\pi _0(\mathring {\mathrm {HM}})$, so that $\Omega$ is not a map of unital monoids.

In fact $\Omega$, as a map of sets, is a right inverse of $\hat \omega$, i.e. $\hat \omega \circ \Omega$ is the identity of $\hat {{\mathcal {Q}}}$. Moreover, $\Omega$ hits all elements of $\pi _0(\mathring {\mathrm {HM}})$ of the form $\pi _0(1,\mathfrak {c}_a)$ and by Lemma 2.18 every element of $\pi _0(\mathring {\mathrm {HM}}_+)$ can be written as a product of one or more elements of the form $\pi _0(1,\mathfrak {c}_a)$. It follows that $\Omega$ is a bijection between $\hat {{\mathcal {Q}}}$ and $\pi _0(\mathring {\mathrm {HM}}_+)$ and this concludes the proof.

2.4 Computation $\pi _0(\breve {\mathrm {HM}})$

We conclude the section by computing $\pi _0(\breve {\mathrm {HM}})$. Recalling Notations 2.5 and 2.6, it suffices to compute $\pi _0(\breve {\mathrm {HM}}_+)$. The canonical structure we have on this last set is that of non-unital monoid, since the multiplication $\mu$ of $\breve {\mathrm {HM}}$ restricts to a map $\breve {\mathrm {HM}}_+\times \breve {\mathrm {HM}}_+\to \breve {\mathrm {HM}}_+$. The total monodromy gives again a morphism of non-unital monoids

\[ \omega\colon\pi_0(\breve{\mathrm{HM}}_+)\to G. \]

In other words, the unital monoid $\pi _0(\breve {\mathrm {HM}})$ is isomorphic to $G\sqcup \{\mathbb {1}\}$, where the extra element $\mathbb {1}$ plays the role of the monoid unit and the old unit $\mathbb {1}_G\in G$ still satisfies $\mathbb {1}_G\cdot g=g\cdot \mathbb {1}_G=g$ for all $g\in G$, but $\mathbb {1}\cdot \mathbb {1}_G=\mathbb {1}_G\cdot \mathbb {1}=\mathbb {1}_G$.

We observe that the hypothesis that $G$ is generated by $\mathfrak {e}({\mathcal {Q}})$ is necessary in Theorem 2.19: if, for instance, ${\mathcal {Q}}=\{\mathbb {1}\}$ and $G$ is any non-trivial group, then $\pi _0(\breve {\mathrm {HM}}_+)$ can rather be identified (as a set) with $G\times G$ and $\omega$ with the product map $G\times G\to G$.

The rough idea of the proof of Theorem 2.19 is the following: given a configuration $(t,\mathfrak {c})$, we can shrink or stretch it until we have $t=1$; we can move points of $\mathfrak {c}$ to either horizontal side of $\breve {{\mathcal {R}}}$, reducing to a configuration $\mathfrak {c}$ supported on $\breve {\partial }$; we can let all points on either component of $\breve {\partial }$ collide with each other, reducing to a configuration $\mathfrak {c}$ supported on at most two points lying on $\breve {\partial }$; finally, we can use that $\mathfrak {e}(Q)$ generates $G$ to ‘trade’ factors of the total monodromy from one component of $\breve {\partial }$ to the other, reaching a configuration $\mathfrak {c}$ supported on a single point.

The rest of the subsection is devoted to the proof of Theorem 2.19. We replace $\breve {\mathrm {HM}}$ by the homotopy equivalent space $\mathrm {Hur}(\breve {{\mathcal {R}}},\breve {\partial })$, see Lemma 2.7.

Notation 2.20 We denote by $\breve {{\mathcal {R}}}^{\mathrm {lr}}$ the horizontally closed square $[0,1]\times (0,1)\subset \mathbb {H}$ and by $\breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}}=\{0,1\}\times (0,1)$ the union of the vertical sides of $\breve {{\mathcal {R}}}^{\mathrm {lr}}$. We abbreviate the nice couple $(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}})$ as $(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$. See Figure 3.

Figure 3. The nice couples $(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$ and $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })$.

We fix once and for all a semialgebraic homeomorphism $\xi ^{\mathrm {rot}}\colon \mathbb {C}\to \mathbb {C}$ which fixes the basepoint $*=-\sqrt {-1}$ and restricts to the homeomorphism $\breve {{\mathcal {R}}}^{\mathrm {lr}}\to \breve {{\mathcal {R}}}$ given by the $90^\circ$ clockwise rotation around $z_{\mathrm {c}}$ (see Notation 2.16).

By functoriality we have a homeomorphism $\xi ^{\mathrm {rot}}_*\colon \mathrm {Hur}(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })\to \mathrm {Hur}(\breve {{\mathcal {R}}},\breve {\partial })$. We will prove Theorem 2.19 by classifying connected components of $\mathrm {Hur}_+(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$; from now on we will focus on the latter space.

To prove Lemma 2.21 we will use the following family of homotopies of $\mathbb {C}$.

Definition 2.22 For all $0< t<1$ we define homotopies ${\mathcal {H}}^{\mathrm {l}}_t,{\mathcal {H}}^{\mathrm {r}}_t\colon \colon \mathbb {C}\times [0,1]\to \mathbb {C}$ by the following formulas:

\begin{gather*} {\mathcal{H}}^{\mathrm{l}}_t(z,s)=\left\{\begin{array}{@{}ll} z & \text{if } \Re(z)\leq0 \text{ or }\Re(z)\geq 1, \\ z-s\Re(z) & \text{if } 0\leq \Re(z)\leq t, \\ z - \Big(\dfrac1{1-st} -1\Big)(1-\Re(z)) & \text{if } t\leq \Re(z)\leq 1; \end{array}\right.\\ {\mathcal{H}}^{\mathrm{r}}_t(z,s)=\left\{\begin{array}{@{}ll} z & \text{if } \Re(z)\leq0 \text{ or }\Re(z)\geq 1, \\ z + \Big(\dfrac1{1-s+st} -1\Big)\Re(z) & \text{if } 0\leq \Re(z)\leq t, \\ z+s(1-\Re(z)) & \text{if } t\leq \Re(z)\leq 1. \end{array}\right. \end{gather*}

Roughly speaking, ${\mathcal {H}}^{\mathrm {l}}_t$ collapses the vertical strip $[0,t]\times \mathbb {R}$ to the vertical line $\mathbb {C}_{\Re =0}$ and expands the vertical strip $[t,1]\times \mathbb {R}$ to the vertical strip $[0,1]\times \mathbb {R}$; similarly ${\mathcal {H}}^{\mathrm {r}}_t$ collapses $[t,1]\times \mathbb {R}$ to $\mathbb {C}_{\Re =1}$ and expands $[0,t]\times \mathbb {R}$ to $[0,1]\times \mathbb {R}$. Both homotopies restrict at each time $s$ to an endomorphism of the nice couple $(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$, so they induce homotopies

\[ ({\mathcal{H}}^{\mathrm{l}}_t)_*,\ ({\mathcal{H}}^{\mathrm{r}}_t)_*\ \colon\mathrm{Hur}(\breve{{\mathcal{R}}}^{\mathrm{lr}},\breve{\partial})\times[0,1]\to\mathrm{Hur}(\breve{{\mathcal{R}}}^{\mathrm{lr}},\breve{\partial}). \]

The following rhombus will help us to define a homotopy of $\mathbb {C}$ that squeezes the two segments in $\breve {\partial }$ to the two central points.

Definition 2.23 We define $\diamond$ as the closed subspace of $\mathbb {H}$ given by

\[ \diamond=\big\{z\in\mathbb{H}\,\colon\,\big|\Re(z)-\tfrac 12\big|+\big|\Im(z)-\tfrac 12\big|\leq\tfrac 12\big\}. \]

Geometrically, $\diamond$ is a closed rhombus centred at the point $z_{\mathrm {c}}$ (see Notation 2.16). The boundary $\partial \diamond$ contains points $z$ for which equality holds in the formula above. The corners of $\diamond$ are denoted by $z_{\diamond }^{\mathrm {l}}=\frac {\sqrt {-1}}2$, $z_{\diamond }^{\mathrm {r}}=1+\frac {\sqrt {-1}}2$, $z_{\diamond }^{\mathrm {u}}=\frac 12+ \sqrt {-1}$ and $z_{\diamond }^{\mathrm {d}}=\frac 12$. We denote by $\breve {\diamond }^{\mathrm {lr}}$ the subspace of $\diamond$ given by

\begin{align*} \breve{\diamond}^{\mathrm{lr}}=(\diamond\setminus\partial\diamond)\cup \{z_{\diamond}^{\mathrm{l}},z_{\diamond}^{\mathrm{r}}\}; \end{align*}

we use the notation $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}=\{z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}\}=\breve {\diamond }^{\mathrm {lr}}\cap \breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}}$ and we abbreviate the nice couple $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial }\breve {\diamond }^{\mathrm {lr}})$ as $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })$; compare with Notation 2.3 and see Figure 3.

We have an inclusion of nice couples $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })\subset (\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$, inducing an inclusion $\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })\subset \mathrm {Hur}(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$.

Before proving Lemma 2.24 we define a suitable homotopy of $\mathbb {C}$.

Definition 2.25 For $z\in \mathbb {C}$ let $\mathfrak {d}^{\diamond }(z)=\min \{|\Re (z)-\frac 12| ;\frac 12\}$. We define a homotopy ${\mathcal {H}}^{\diamond }\colon \mathbb {C}\times [0,1]\to \mathbb {C}$ by the following formula:

\[ {\mathcal{H}}^{\diamond}(z,s)=\left\{\begin{array}{@{}ll} z-s \mathfrak{d}^{\diamond}(z)\sqrt{-1} & \text{if } \Im(z)\geq 1,\\ z- 2s\mathfrak{d}^{\diamond}(z)\left(\Im(z)-\dfrac 12\right)\sqrt{-1} & \text{if } 0 \leq \Im(z)\leq 1,\\ z+s\Big(\dfrac{\Im(z)}2 +\mathfrak{d}^\diamond(z)\Big)\sqrt{-1} & \text{if } \Im(z)\leq 0. \end{array}\right. \]

The homotopy ${\mathcal {H}}^{\diamond }$ satisfies the following properties:

• • for all $0\leq s\leq 1$, the map ${\mathcal {H}}^{\diamond }(-;s)\colon \mathbb {C}\to \mathbb {C}$ induces an endomorphism of the nice couple $(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$ and an endomorphism of the nice couple $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })$;

• • ${\mathcal {H}}^{\diamond }(-;0)$ is the identity of $\mathbb {C}$;

• • ${\mathcal {H}}^{\diamond }(-;1)$ sends $\breve {{\mathcal {R}}}^{\mathrm {lr}}$ onto $\breve {\diamond }^{\mathrm {lr}}$ and $\breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}}$ onto $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}$.

Note also that if $\mathfrak {c}\in \mathrm {Hur}(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$ is supported in $\breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}}$, then the entire path ${\mathcal {H}}^{\diamond }_*(\mathfrak {c},-)$ consists of configurations supported in $\breve {\partial }\breve {{\mathcal {R}}}^{\mathrm {lr}}$. Using Lemmas 2.21 and 2.24 together, we can therefore connect any $\mathfrak {c}\in \mathrm {Hur}(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$ to a configuration $\mathfrak {c}'\in \mathrm {Hur}(\breve {{\mathcal {R}}}^{\mathrm {lr}},\breve {\partial })$ supported in $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}=\{z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}\}$.

We next define auxiliary configurations, supported on the three points $z_{\mathrm {c}},z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}$: by moving $z_{\mathrm {c}}$ towards $z_{\diamond }^{\mathrm {l}}$ or towards $z_{\diamond }^{\mathrm {r}}$, we can construct paths between configurations supported on $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}=\{z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}\}$.

3.1 Bar constructions

We recall the classical definition of bar construction with respect to a topological monoid $M$ and a left $M$-module $X$.

In fact Definition 3.1 only uses that $M$ is an associative non-unital monoid; in § 3.3 we will recall the thin bar construction, which is a simplicial space whose degeneracy maps are defined using the unit $e\in M$.

It is a standard fact that if $M$ is a topological monoid, $X$ is a left $M$-module and for all $m\in M$ the map $\mu (m;-)\colon X\to X$ is a self-homotopy equivalence of $X$, then the natural projection map $p_X\colon B(M,X)\to BM=B(M,*)$ induced by the constant, $M$-equivariant map $X\to *$ is a quasi-fibration with fibres homeomorphic to $X$. See, for instance, [Reference HatcherHat14, Lemma D.1].

Lemma 3.2 Recall Definitions 2.4 and 2.12 and let $(t,\mathfrak {c})\in \breve {\mathrm {HM}}$; then the left multiplication $\mu ((t,\mathfrak {c}),-)$ restricts to a self-homotopy equivalence of $\breve {\mathrm {HM}}_+$; moreover, if $\omega (\mathfrak {c})=\mathbb {1}\in G$, then $\mu ((t,\mathfrak {c}),-)|_{\breve {\mathrm {HM}}_+}$ is homotopic to the identity of $\breve {\mathrm {HM}}_+$. It follows that

\[ p_{\breve{\mathrm{HM}}_+}\colon B(\breve{\mathrm{HM}},\breve{\mathrm{HM}}_+)\to B\breve{\mathrm{HM}} \]

is a quasifibration with fibre $\breve {\mathrm {HM}}_+$.

Proof. It suffices to prove the statement for one configuration $(t,\mathfrak {c})$ in each connected component of $\breve {\mathrm {HM}}$: the statement is obvious for $(t,\mathfrak {c})=(0,(\emptyset, \mathbb {1},\mathbb {1}))\in \breve {\mathrm {HM}}$, which is the neutral element of $\breve {\mathrm {HM}}$. Using Theorem 2.19 we can then assume that $t=1$ and $\mathfrak {c}$ has the form $\mathfrak {c}^{\mathrm {d}}_g:=(P,\psi,\varphi )$ for some $g\in G$, where:

• • $P=\{z_{\diamond }^{\mathrm {d}}\}$ consists of the only point $z_{\diamond }^{\mathrm {d}}$ (see Definition 2.23);

• • $\psi \colon \mathfrak {Q}(P)=\{\mathbb {1}\}\to {\mathcal {Q}}$ is the trivial map of PMQs;

• • $\varphi \colon \mathfrak {G}(P)\to G$ sends the unique standard generator of $\mathfrak {G}(P)$ to $g$.

We start with the case $g=\mathbb {1}$. We claim that $\mu ((1,\mathfrak {c}^{\mathrm {d}}_{\mathbb {1}}),-)|_{\breve {\mathrm {HM}}_+}$ is homotopic to the identity of $\breve {\mathrm {HM}}_+$; by Lemma 2.7 it suffices to prove that the restriction

\[ \mu((1,\mathfrak{c}_{\mathbb {1}}^{\mathrm{d}}),-)\colon\mathrm{Hur}_+(\breve{{\mathcal{R}}},\breve{\partial})\to\breve{\mathrm{HM}}_+ \]

is homotopic to the natural inclusion $\mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\hookrightarrow \breve {\mathrm {HM}}_+$.

First we prove that the maps $\mu ((1,\mathfrak {c}_{\mathbb {1}}^{\mathrm {d}}),-)$ and $\mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)$ are homotopic maps $\mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$. We use an argument similar to the proof [Reference BianchiBia23a, Proposition 7.10]. Recall from [Reference BianchiBia23a, Definition 3.1] that the Ran space $\mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)$ is the space of non-empty finite subsets of $\breve {{\mathcal {R}}}_2$; it is weakly contractible [Reference LurieLur17, Theorem 5.5.1.6] and using the notion of standard explosion from [Reference BianchiBia23a, Subsection 7.2], one can find a homotopy $\mathscr {E}^{z_{\diamond }^{\mathrm {d}}}\colon \mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)\times [0,1]\to \mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)$ contracting $\mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)$ onto the configuration $\{z_{\diamond }^{\mathrm {d}}\}$. Recall also that there is an external product $-\times -\colon \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })\times \mathrm {Ran}_+(\breve {{\mathcal {R}}}_2) \to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$, which essentially superposes to a configuration in $\mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$ another configuration with trivial monodromies (i.e. a configuration in $\mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)$); see [Reference BianchiBia23a, Definition 5.7 and Notation 5.9].

We consider the following homotopy ${\mathcal {H}}^{z_{\diamond }^{\mathrm {d}}}\colon \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })\times [0,1]\to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$

where $\varepsilon \colon \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })\to \mathrm {Ran}_+(\breve {{\mathcal {R}}}_2)$ is the canonical map $(P,\psi,\varphi )\mapsto P$. Roughly speaking, each point in the support of a configuration in $\mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$ is split at time 0 into two points: the first keeps the original local monodromy and does not move; the second carries a trivial local monodromy and moves straightly to $z_{\diamond }^{\mathrm {d}}$; at time 1 all the second points have merged at $z_{\diamond }^{\mathrm {d}}$. We observe the following:

• • the composition ${\mathcal {H}}^{z_{\diamond }^{\mathrm {d}}}(-,0)\circ \mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)\colon \mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$ is equal to $\mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)$, since ${\mathcal {H}}^{z_{\diamond }^{\mathrm {d}}}(-,0)$ is the identity of $\mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$;

• • the composition ${\mathcal {H}}^{z_{\diamond }^{\mathrm {d}}}(-,1)\circ \mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)\colon \mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })$ is equal to $\mu ((1,\mathfrak {c}_{\mathbb {1}}^{\mathrm {d}}),-)$.

We thus obtain that $\mu ((1,\mathfrak {c}_{\mathbb {1}}^{\mathrm {d}}),-)$ and $\mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)$ are homotopic as maps $\mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\to \mathrm {Hur}_+(\breve {{\mathcal {R}}}_2,\breve {\partial })\subset \breve {\mathrm {HM}}_+$.

We then note that $(1,(\emptyset,\mathbb {1},\mathbb {1}))$ is connected by a path to $(0,(\emptyset,\mathbb {1},\mathbb {1}))$ in $\breve {\mathrm {HM}}$; as a consequence $\mu ((1,(\emptyset,\mathbb {1},\mathbb {1})),-)$ and $\mu ((0,(\emptyset,\mathbb {1},\mathbb {1})),-)$ are homotopic as maps $\mathrm {Hur}_+(\breve {{\mathcal {R}}},\breve {\partial })\to \breve {\mathrm {HM}}_+$ and the second map is the natural inclusion. This concludes the case $g=\mathbb {1}$.

Now let $g\neq \mathbb {1}$; by Theorem 2.19 the three configurations $\mu ((1,\mathfrak {c}_g^{\mathrm {d}}),(1,\mathfrak {c}_{g^{-1}}^{\mathrm {d}}))$, $\mu ((1,\mathfrak {c}_{g^{-1}}^{\mathrm {d}}), (1,\mathfrak {c}_g^{\mathrm {d}}))$ and $(1,\mathfrak {c}_{\mathbb {1}}^{\mathrm {d}})$ are in the same connected component of $\breve {\mathrm {HM}}_+$: hence, $\mu ((1,\mathfrak {c}_g^{\mathrm {d}}),-)$ and $\mu ((1,\mathfrak {c}_{g^{-1}}^{\mathrm {d}}),-)$ are homotopy inverses as maps $\breve {\mathrm {HM}}_+\to \breve {\mathrm {HM}}_+$.

It is a classical fact that if $M$ is a unital, topological monoid, then $EM$ is contractible. In the following proposition we prove an analogous statement for $B(\breve {\mathrm {HM}},\breve {\mathrm {HM}}_+)$.

The proof of Proposition 3.3 is in Appendix A.3. As a consequence of Lemma 3.2 and Proposition 3.3 we obtain the following theorem.

3.2 Pontryagin ring and group completion

If $M$ is a unital topological monoid, $H_*(M)$ is an associative, graded ring with unit, called a Pontryagin ring. We usually denote by $x\cdot y\in H_*(M)$ the Pontryagin product of two homology classes $x,y\in H_*(M)$. The unit $1\in H_0(M)$ is the homology class corresponding to the connected component of $e$ in $\pi _0(M)$. The subset $\pi _0(M)\subset H_0(M)\subset H_*(M)$ is closed under multiplication.

Note that if $M$ is weakly braided, then the ring localisation $H_*(M)[\pi _0(M)^{-1}]$ can be constructed by right fractions: for all $x\in H_*(M)$ and $a\in \pi _0(M)$ there exist $y\in H_*(M)$ and $b\in \pi _0(M)$ with $x\cdot b=a\cdot y$. This follows from setting $b=a$ and $y=(p_2)_*\circ \mathfrak {br}_*(x\times a)$, where $\times$ denotes the homology cross-product and $p_2\colon M\times M\to M$ is as in Definition 3.5.

Proof. We define $\mathfrak {br}\colon \mathring {\mathrm {HM}}\times \mathring {\mathrm {HM}}\to \mathring {\mathrm {HM}}\times \mathring {\mathrm {HM}}$ by the formula

\[ \mathfrak{br}((t,\mathfrak{c}),(t',\mathfrak{c}'))=((t',\mathfrak{c}'),(t,\mathfrak{c}^{\hat\omega(\mathfrak{c}')})), \]

where $\hat \omega$ is the $\hat {{\mathcal {Q}}}$-valued total monodromy (see Notation 2.14) and we use the action by global conjugation [Reference BianchiBia23a, Definition 6.6]. It is clear that $\mathfrak {br}$ is a homeomorphism and that the first property in Definition 3.5 holds. To check the second property, note that $\mathfrak {br}$ restricts to a map

\[ \mathfrak{br}\colon \mathrm{Hur}(\mathring{{\mathcal{R}}})\times\mathrm{Hur}(\mathring{{\mathcal{R}}})\to \mathrm{Hur}(\mathring{{\mathcal{R}}})\times\mathrm{Hur}(\mathring{{\mathcal{R}}}). \]

By Lemma 2.7 it suffices to prove that $\mu$ and $\mu \circ \mathfrak {br}$ are homotopic when considered as maps $\mathrm {Hur}(\mathring {{\mathcal {R}}})\times \mathrm {Hur}(\mathring {{\mathcal {R}}})\to \mathrm {Hur}(\mathring {{\mathcal {R}}}_2)$. Let $\mathring {{\mathcal {R}}}^{1/2}\subset \mathring {{\mathcal {R}}}$ be the open unit square $(1/4,3/4)\times (1/4,3/4)$ of side length $1/2$ centred at $z_{\mathrm {c}}\in \mathring {{\mathcal {R}}}$; we can regard $\mathrm {Hur}(\mathring {{\mathcal {R}}}^{1/2})$ as an open subspace of $\mathrm {Hur}(\mathring {{\mathcal {R}}})$, containing all configurations supported in $\mathring {{\mathcal {R}}}^{1/2}$. Note that $\mathfrak {br}$ restricts to a map

\[ \mathfrak{br}\colon \mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2})\times\mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2})\to \mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2})\times\mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2}). \]

Let ${\mathcal {H}}^{1/2}\colon \mathbb {C}\times [0,1]\to \mathbb {C}$ be a semialgebraic isotopy of $\mathbb {C}$ fixing $*$ at all times, such that ${\mathcal {H}}^{1/2}(-,0)=\operatorname {Id}_{\mathbb {C}}$ and ${\mathcal {H}}^{1/2}(-,1)$ restricts to a homeomorphism $\mathring {{\mathcal {R}}}\to \mathring {{\mathcal {R}}}^{1/2}$. Then by functoriality there is a deformation of $\mathrm {Hur}(\mathring {{\mathcal {R}}})$ into the subspace $\mathrm {Hur}(\mathring {{\mathcal {R}}}^{1/2})$. Thus, it suffices to prove that the following restricted maps are homotopic:

\[ \mu,\ \mu\circ\mathfrak{br}\ \colon\mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2})\times\mathrm{Hur}(\mathring{{\mathcal{R}}}^{1/2})\to\mathrm{Hur}(\mathring{{\mathcal{R}}}_2). \]

Let ${\mathcal {H}}^{\mathfrak {br}}\colon \mathbb {C}\times [0,1]\to \mathbb {C}$ be a semialgebraic isotopy of $\mathbb {C}$ fixing pointwise $\mathbb {C}\setminus \mathring {{\mathcal {R}}}_2$ at all times, such that ${\mathcal {H}}^{\mathfrak {br}}(-,0)=\operatorname {Id}_{\mathbb {C}}$ and ${\mathcal {H}}^{\mathfrak {br}}(-,1)\colon \mathbb {C}\to \mathbb {C}$ has the following properties:

• • ${\mathcal {H}}^{\mathfrak {br}}(-,1)$ restricts to $\tau _1\colon \mathring {{\mathcal {R}}}^{1/2}\to \tau _1(\mathring {{\mathcal {R}}}^{1/2})$ (see Definition 2.9);

• • ${\mathcal {H}}^{\mathfrak {br}}(-,1)$ restricts to $\tau _{-1}\colon \tau _1(\mathring {{\mathcal {R}}}^{1/2})\to \mathring {{\mathcal {R}}}^{1/2}$;

• • ${\mathcal {H}}^{\mathfrak {br}}(-,1)$ restricts to a self-homeomorphism of $\mathbb {C}\setminus (\mathring {{\mathcal {R}}}^{1/2}\cup \tau _1(\mathring {{\mathcal {R}}}^{1/2}))$ representing a clockwise half Dehn twist, for instance we may assume that there is a simple loop $\gamma \subset \mathbb {S}_{1,2}\setminus \tau _1(\mathring {{\mathcal {R}}}^{1/2})$ spinning clockwise around $\tau _1(\mathring {{\mathcal {R}}}^{1/2})$, such that ${\mathcal {H}}^{\mathfrak {br}}(-,1)\circ \gamma$ is a simple loop contained in $\mathbb {S}_{0,1}\setminus \mathring {{\mathcal {R}}}^{1/2}$ and spinning clockwise around $\mathring {{\mathcal {R}}}^{1/2}$.

Then the composition of $\mu$ with ${\mathcal {H}}^{\mathfrak {br}}_*$ gives a homotopy from $\mu$ to $\mu \circ \mathfrak {br}$ as maps $\mathrm {Hur}(\mathring {{\mathcal {R}}}^{1/2})\times \mathrm {Hur}(\mathring {{\mathcal {R}}}^{1/2})\to \mathrm {Hur}(\mathring {{\mathcal {R}}}_2)$.

Recall that for any topological monoid $M$ there is a canonical map $M\to \Omega BM$: the induced map in homology $H_*(M)\to H_*(\Omega BM)$ sends the multiplicative subset $\pi _0(M)\subset H_*(M)$ to the set of invertible elements of the Pontryagin ring $H_*(\Omega BM)$. Therefore, there is an induced map of rings

\begin{align*} H_*(M)[\pi_0(M)^{-1}]\to H_*(\Omega BM). \end{align*}

We recall the group completion theorem (see [Reference McDuff and SegalMS76] and [Reference Friedlander and MazurFM94, Theorem Q.4]).

Theorem 3.7 (Group completion theorem)

Let $M$ be a topological monoid and suppose that the localisation $H_*(M)[\pi _0(M)^{-1}]$ can be constructed by right fractions. Then the canonical map

\[ H_*(M)[\pi_0(M)^{-1}]\to H_*(\Omega BM) \]

is an isomorphism of rings.

Using Theorem 3.7 together with Lemma 3.6 we obtain an isomorphism of rings

\[ H_*(\mathring{\mathrm{HM}})[\pi_0(\mathring{\mathrm{HM}})^{-1}]\cong H_*(\Omega B\mathring{\mathrm{HM}}). \]

3.3 Thin bar construction

Recall Definition 3.1: the semisimplicial space $B_{\bullet } (M,X)$ can be enhanced to a simplicial space by defining the degeneracy map $s_i\colon B_k(M,X)\to B_{k+1}(M,X)$, for $0\leq i\leq k$, by the following formula, where $e$ denotes the neutral element of $M$,

\[ s_i\colon(m_1,\ldots,m_k,x)\mapsto (m_1,\ldots,m_i,e,m_{i+1},\ldots,m_k,x). \]

It is a classical fact that if $M$ is well-pointed, then $p_{\bar {B}}\colon BM\to \bar {B} M$ is a weak homotopy equivalence, as $\bar {B}_{\bullet }(M)$ is a good simplicial space in the sense of [Reference SegalSeg73, Appendix 2]. The monoids $\mathring {\mathrm {HM}}$ and $\breve {\mathrm {HM}}$ are well-pointed, as the connected component of the unit $(0,(\emptyset,\mathbb {1},\mathbb {1}))$ is contractible in both cases.

4.1 Definition of the comparison map

By Definition 3.1 the space $B\mathring {\mathrm {HM}}$ (respectively, $B\breve {\mathrm {HM}}$) arises as a quotient of the disjoint union $\coprod _{p\geq 0}\Delta ^p\times \mathring {\mathrm {HM}}^p$ (respectively, $\coprod _{p\geq 0}\Delta ^p\times \breve {\mathrm {HM}}^p$). We will first define $\sigma$ on this disjoint union and then prove that the given assignment induces a map on the quotient.

Given $(\underline {w},\underline {t},\underline {\mathfrak {c}})$ as in Notation 4.2, note that the product $(t_1,\mathfrak {c}_1)\cdots (t_p,\mathfrak {c}_p)$ has the form $(t_1+\cdots +t_p,\mathfrak {c})$, with $\mathfrak {c}$ supported on the set

\[ P=P_1\cup(P_2+t_1)\cup(P_3+t_1+t_2)\cup\cdots\cup(P_p+t_1+\cdots+t_{p-1}). \]

By Definition 2.4 we have $\mathfrak {c}\in \mathrm {Hur}(\mathring {{\mathcal {R}}}_{\infty })$ (respectively, $\mathfrak {c}\in \mathrm {Hur}(\breve {{\mathcal {R}}}_{\infty },\breve {\partial })$), but in the following we will consider $\mathfrak {c}$ as a configuration in $\mathrm {Hur}(\mathring {{\mathcal {R}}}_{\mathbb {R}})$ (in $\mathrm {Hur}(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })$, see Notation 2.3).

Definition 4.3 The above assignment $(\underline {w};\underline {t},\underline {\mathfrak {c}})\mapsto \mathfrak {c}$ gives a continuous map

\[ \hat{\mu}\colon \coprod_{p\geq 0} \Delta^p\times(\mathring{\mathrm{HM}})^p \to \mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}}) \quad \bigg( \text{respectively, } \hat{\mu}\colon \coprod_{p\geq 0} \Delta^p\times(\breve{\mathrm{HM}})^p \to \mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{\partial}) \bigg). \]

Note that $\hat {\mu }$ factors, on each subspace $\Delta ^p\times (\mathring {\mathrm {HM}})^p$ (respectively, $\Delta ^p\times (\breve {\mathrm {HM}})^p$), through the projection on the factor $(\mathring {\mathrm {HM}})^p$ (respectively, $(\breve {\mathrm {HM}})^p$). The first subspace $\Delta ^0$ is sent to the empty product in $\mathring {\mathrm {HM}}$ (in $\breve {\mathrm {HM}}$), i.e. to the neutral element $(0,(\emptyset,\mathbb {1},\mathbb {1}))$.

See Figure 5(left). Note that the barycentres $b,b^+,b^-$ vary continuously on $\coprod _{p\geq 0} \Delta ^p\times (\mathring {\mathrm {HM}})^p$ (on $\coprod _{p\geq 0} \Delta ^p\times (\breve {\mathrm {HM}})^p$), but do not factor to continuous functions on $B\mathring {\mathrm {HM}}$ (respectively, $B\breve {\mathrm {HM}}$): indeed, if $w_0=0$, the triple $(\underline {w};\underline {t},\underline {\mathfrak {c}})$ is equivalent to the triple $(\underline {w}';\underline {t}',\underline {\mathfrak {c}}')$ obtained by removing $w_0$, $t_1$ and $\underline {\mathfrak {c}}_1$; all barycentres $b,b^+,b^-$ drop by $t_1$ when passing from the first to the second triple. Nevertheless, the differences $b^+-b$ and $b-b^-$ factor to continuous functions defined on $B\mathring {\mathrm {HM}}$ (respectively, $B\breve {\mathrm {HM}}$).

Note also that for all $(\underline {w};\underline {t},\underline {\mathfrak {c}})$ we have $a_0\leq b^-\leq b\leq b^+\leq a_p$. More precisely, let $i_{\rm min}\ge 0$ be minimal with $w_{i_{\rm min}}>0$ and let $i_{\rm max}\le p$ be maximal with $w_{i_{\rm max}}>0$; then $a_{i_{\rm min}}\leq b^-\leq b\leq b^+\leq a_{i_{\rm max}}$, with all these inequalities strict unless they are all equalities: in this case all $t_i$ with $i_{\rm min}< i\le i_{\rm max}$ are equal to $0$ and all corresponding $\mathfrak {c}_i$ are equal to $(\emptyset,\mathbb {1},\mathbb {1})$, so that $\hat {\mu }(\underline {w},\underline {t},\underline {\mathfrak {c}})$ is equal to the product $(t_1,\mathfrak {c}_1)\cdots (t_{i_{\rm min}},\mathfrak {c}_{i_{\rm min}})\cdot (t_{i_{\rm max}+1},\mathfrak {c}_{i_{\rm max}+1})\cdots (t_p,\mathfrak {c}_p)$. In particular, if $b^-=b=b^+$, then we have that $\hat {\mu }(\underline {w},\underline {t},\underline {\mathfrak {c}})$ is supported away from $\mathbb {S}_{b,b}$, in fact it is supported away from $\mathbb {S}_{b-\varepsilon,b+\varepsilon }$ for $\varepsilon >0$ small enough.

Definition 4.5 We define a continuous function $\varepsilon \colon \coprod _{p\geq 0} \Delta ^p\times (\mathring {\mathrm {HM}})^p\to [0,1]$ (respectively, $\varepsilon \colon \coprod _{p\geq 0} \Delta ^p\times (\breve {\mathrm {HM}})^p\to [0,1]$): for $(\underline {w};\underline {t},\underline {\mathfrak {c}})$ as in Notation 4.2, we denote by $P\subset \mathbb {R}\times [0,1]$ the support of $\hat {\mu }(\underline {w};\underline {t},\underline {\mathfrak {c}})$ and set

\[ \varepsilon\colon(\underline{w};\underline{t},\underline{\mathfrak{c}})=\tfrac 12\sup\{t\in[0,1]\,|\,P\cap\mathbb{S}_{b^--t,b^++t}=\emptyset\}, \]

where upper and lower barycentres are computed with respect to $(\underline {w};\underline {t},\underline {\mathfrak {c}})$. We denote $b^-_\varepsilon =b^--\varepsilon$ and $b^+_\varepsilon =b^++\varepsilon$.

We observe that $\varepsilon$ satisfies the following properties:

• • for all $(\underline {w},\underline {t},\underline {\mathfrak {c}})$ satisfying $b^-=b^+$, we have $\varepsilon (\underline {w},\underline {t},\underline {\mathfrak {c}})>0$;

• • for all $(\underline {w},\underline {t},\underline {\mathfrak {c}})$ with $\varepsilon (\underline {w},\underline {t},\underline {\mathfrak {c}})>0$, the configuration $\hat {\mu }(\underline {w},\underline {t},\underline {\mathfrak {c}})$ is supported away from $\mathbb {S}_{b^--\varepsilon,b^++\varepsilon }$.

The advantage of replacing $b^-$ and $b^+$ by $b^-_\varepsilon$ and $b^+_\varepsilon$ is that we now have strict inequalities $b^-_\varepsilon < b< b^+_\varepsilon$ for all $(\underline {w};\underline {t},\underline {\mathfrak {c}})$. As we will see, the disadvantage that $b^+_\varepsilon -b^-_\varepsilon$ does not factor through a function defined on $B\mathring {\mathrm {HM}}$ (respectively, on $B\breve {\mathrm {HM}}$) will be inessential.

Recall the proof of Lemma 2.7: for $s>0$ the map $\Lambda _s\colon \mathbb {C}\to \mathbb {C}$ is an endomorphism of the nice couple $(\mathring {{\mathcal {R}}}_{\mathbb {R}},\emptyset )$ (respectively, $(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })$), depending continuously on $s$. We obtain a continuous map

\[ \Lambda_* \!\colon\! \mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}})\times (0,\infty)\to\!\mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}}) \quad (\!\text{respectively, }\Lambda_*\! \colon\! \mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{\partial})\times (0,\infty)\to\!\mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{\partial})\!). \]

Similarly, recall Definition 2.9: for all $t\in \mathbb {R}$ the map $\tau _t$ is an endomorphism of the nice couple $(\mathring {{\mathcal {R}}}_{\mathbb {R}},\emptyset )$ (respectively, $(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })$), depending continuously on $t$. We obtain a continuous map

\[ \tau_* \colon \mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}})\times \mathbb{R} \to\mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}})\quad\quad (\text{respectively, }\tau_* \colon \mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{\partial})\times \mathbb{R} \to\mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{\partial})). \]

Definition 4.6 We define a map

\[ \hat{\mu}^b\colon \coprod_{p\geq 0} \Delta^p\times(\mathring{\mathrm{HM}})^p \to \mathrm{Hur}(\mathring{{\mathcal{R}}}_{\mathbb{R}})\quad \bigg(\text{respectively, } \hat{\mu}^b\colon \coprod_{p\geq 0} \Delta^p\times(\breve{\mathrm{HM}})^p \to \mathrm{Hur}(\breve{{\mathcal{R}}}_{\mathbb{R}})\bigg) \]

by the following assignment:

\[ (\underline{w};\underline{t},\underline{\mathfrak{c}})\mapsto \Lambda_*\bigg(\tau_*(\hat{\mu}(\underline{w};\underline{t},\underline{\mathfrak{c}}); -b^-_\varepsilon(\underline{w};\underline{t},\underline{\mathfrak{c}})); \frac{1}{b^+_\varepsilon(\underline{w};\underline{t},\underline{\mathfrak{c}})-b^-_\varepsilon(\underline{w};\underline{t},\underline{\mathfrak{c}})}\bigg). \]

Roughly speaking, the map $\hat {\mu }^b$ has the effect of a horizontal translation and a dilation of the configuration $\hat {\mu }(\underline {w};\underline {t},\underline {\mathfrak {c}})$: the effect of the translation and dilation is to map the rectangle $[b^-_\varepsilon,b^+_\varepsilon ]\times [0,1]$ homeomorphically onto the unit square ${\mathcal {R}}$.

Definition 4.7 Let $\mathring {\diamond }$ denote the interior of $\diamond$ (see Definition 2.23) and denote $\mathbb {R}+ {\sqrt {-1}}/{2}=\{t+ {\sqrt {-1}}/{2}\,|\,t\in \mathbb {R}\}\subset \mathbb {H}$. We introduce several nice couples:

\[ \begin{array}{ll} \bullet\ \breve{\mathfrak{C}}^{\Box}=(\mathring{{\mathcal{R}}}_{\mathbb{R}},\mathring{{\mathcal{R}}}_{\mathbb{R}}\setminus\mathring{{\mathcal{R}}}); & \bullet\ \breve{\mathfrak{C}}^{\diamond}=\bigg(\breve{\diamond}^{\mathrm{lr}}\cup \left(\mathbb{R}+\dfrac{\sqrt{-1}}{2}\right),\left(\breve{\diamond}^{\mathrm{lr}}\cup \left(\mathbb{R}+\dfrac{\sqrt{-1}}{2}\right)\right) \setminus\mathring{\diamond}\bigg);\\ \bullet\ {\mathfrak{C}}^{\Box}=(\breve{{\mathcal{R}}}_{\mathbb{R}},\breve{{\mathcal{R}}}_{\mathbb{R}}\setminus\mathring{{\mathcal{R}}}); & \bullet\ {\mathfrak{C}}^{\diamond}= \bigg(\diamond\cup \left(\mathbb{R}+\dfrac{\sqrt{-1}}{2}\right),\left(\diamond\cup \left(\mathbb{R}+\dfrac{\sqrt{-1}}{2}\right)\right) \setminus\mathring{\diamond}\bigg). \end{array} \]

Since $\operatorname {Id}_\mathbb {C}$ is a map of nice couples $(\mathring {{\mathcal {R}}}_{\mathbb {R}},\emptyset )\to \breve {\mathfrak {C}}^{\Box }$ (respectively, $(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })\to {\mathfrak {C}}^{\Box }$), it induces a map $\mathrm {Hur}(\mathring {{\mathcal {R}}}_{\mathbb {R}})\to \mathrm {Hur}(\breve {\mathfrak {C}}^{\Box })$ (respectively, $\mathrm {Hur}(\breve {{\mathcal {R}}}_{\mathbb {R}},\breve {\partial })\to \mathrm {Hur}({\mathfrak {C}}^{\Box })$). See Figure 4.

Notation 4.8 By abuse of notation we will also denote by $\hat {\mu }^b$ the composition

Figure 4. Top: nice couples $\breve {\mathfrak {C}}^{\Box }$ and $\breve {\mathfrak {C}}^{\diamond }$. Bottom: nice couples ${\mathfrak {C}}^{\Box }$ and ${\mathfrak {C}}^{\diamond }$.

Definition 4.9 We define maps $\kappa ^-$ and $\kappa ^+ \colon \mathbb {C}\times [0,\infty )\to \mathbb {C}$ by the formulas

\begin{gather*} \kappa^-(z,s)= \begin{cases} z & \text{if } \Re(z)\geq 0,\\ z-\Re(z) & \text{if } -s\leq \Re(z)\leq 0,\\ z+s & \text{if }\Re(z)\leq -s ; \end{cases}\\ \kappa^+(z,s)=\begin{cases} z & \text{if } \Re(z)\leq 1,\\ z- \Re(z)+1 & \text{if } 1\leq \Re(z)\leq 1+s,\\ z-s & \text{if }\Re(z)\geq 1+s . \end{cases}\end{gather*}

Roughly speaking, both $\kappa ^-$ and $\kappa ^+$ fix the vertical strip $[0,1]\times \mathbb {R}$ for all $s\geq 0$; the map $\kappa ^-(-,s)$ collapses the strip $[-s,0]\times \mathbb {R}$ to the vertical line $\mathbb {C}_{\Re =0}$ and translates $(-\infty,s]\times \mathbb {R}$ to the right; instead $\kappa ^+(-,s)$ collapses the strip $[1,1+s]\times \mathbb {R}$ to the vertical line $\mathbb {C}_{\Re =1}$ and translates $[1+s,\infty )\times \mathbb {R}$ to the left.

Both $\kappa ^-(-,s)$ and $\kappa ^+(-,s)$ are morphisms of nice couples $\breve {\mathfrak {C}}^{\Box }\to \breve {\mathfrak {C}}^{\Box }$ (respectively, ${\mathfrak {C}}^{\Box }\to {\mathfrak {C}}^{\Box }$) for all $s\geq 0$. We obtain continuous maps

\begin{gather*} \kappa^-_*,\,\kappa^+_*\,\colon\mathrm{Hur}(\breve{\mathfrak{C}}^{\Box})\times[0,\infty)\to \mathrm{Hur}(\breve{\mathfrak{C}}^{\Box})\\ (\text{respectively, }\kappa^-_*,\,\kappa^+_*\,\colon\mathrm{Hur}({\mathfrak{C}}^{\Box})\times[0,\infty)\to \mathrm{Hur}({\mathfrak{C}}^{\Box})). \end{gather*}

Note that ${\mathcal {H}}^{\diamond }_1$ is a morphism of nice couples $\breve {\mathfrak {C}}^{\Box }\to \breve {\mathfrak {C}}^{\diamond }$ (respectively, ${\mathfrak {C}}^{\Box }\to {\mathfrak {C}}^{\diamond }$).

Definition 4.11 We define a map

\[ \hat{\mu}^\Box\colon \coprod_{p\geq 0} \Delta^p\times(\mathring{\mathrm{HM}})^p \to \mathrm{Hur}(\breve{\mathfrak{C}}^{\Box})\quad \bigg(\text{respectively, }\hat{\mu}^\Box\colon \coprod_{p\geq 0} \Delta^p\times(\breve{\mathrm{HM}})^p \to \mathrm{Hur}({\mathfrak{C}}^{\Box})\bigg) \]

by the following assignment, where $p$, $a_p$, $b^-_\varepsilon$ and $b^+_\varepsilon$ depend on $(\underline {w};\underline {t},\underline {\mathfrak {c}})$:

\[ \hat{\mu}^\Box(\underline{w};\underline{t},\underline{\mathfrak{c}})= \kappa^-_*\bigg( \kappa^+_*\bigg( \hat{\mu}^b(\underline{w};\underline{t},\underline{\mathfrak{c}}) ; \frac{a_p-b^+_\varepsilon}{b^+_\varepsilon-b^-_\varepsilon} \bigg) ;\, \frac{b^-_\varepsilon-a_0}{b^+_\varepsilon-b^-_\varepsilon} \bigg). \]

We further define

\[ \hat{\mu}^\diamond\colon \coprod_{p\geq 0} \Delta^p\times(\mathring{\mathrm{HM}})^p \to \mathrm{Hur}(\breve{\mathfrak{C}}^{\diamond})\quad \bigg(\text{respectively, }\hat{\mu}^\diamond\colon \coprod_{p\geq 0} \Delta^p\times(\breve{\mathrm{HM}})^p \to \mathrm{Hur}({\mathfrak{C}}^{\diamond})\bigg) \]

as the composition $({\mathcal {H}}^{\diamond }_1)_*\circ \hat {\mu }^\Box$.

Roughly speaking, $\hat {\mu }^\Box$ improves the effect of $\hat {\mu }^b$ as follows: $\hat {\mu }^b(\underline {w};\underline {t},\underline {\mathfrak {c}})$ is a configuration supported in the rectangle $[ ({b_\varepsilon ^--a_0})/({b_\varepsilon ^+-b_\varepsilon ^-}),1+ ({a_p-b_\varepsilon ^+})/({b_\varepsilon ^+-b_\varepsilon ^-})]\times [0,1]$ and the further application of $\kappa ^-_*$ and $\kappa ^+_*$ collapse $\hat {\mu }^b(\underline {w};\underline {t},\underline {\mathfrak {c}})$ to a configuration supported in ${\mathcal {R}}$. The further composition $\hat {\mu }^\diamond$ changes the configuration $\hat {\mu }^\Box (\underline {w};\underline {t},\underline {\mathfrak {c}})$ to a configuration $\hat {\mu }^\diamond (\underline {w};\underline {t},\underline {\mathfrak {c}})$ supported in $\diamond$.

Note that there is a natural inclusion of spaces $\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })\subset \mathrm {Hur}(\breve {\mathfrak {C}}^{\diamond })$ (respectively, $\mathrm {Hur}(\diamond,\partial )\subset \mathrm {Hur}({\mathfrak {C}}^{\diamond })$). The following lemma summarises the previous discussion.

We consider now the external product

\begin{gather*} -\times-\colon \mathrm{Hur}(\breve{\diamond}^{\mathrm{lr}},\breve{\partial})\times\mathrm{Ran}(\breve{\diamond}^{\mathrm{lr}})\to \mathrm{Hur}(\breve{\diamond}^{\mathrm{lr}},\breve{\partial})\\ (\text{respectively, }-\times-\colon \mathrm{Hur}(\diamond,\partial)\times\mathrm{Ran}(\diamond)\to \mathrm{Hur}(\diamond,\partial)) \end{gather*}

from [Reference BianchiBia23a, Definition 5.7 and Notation 5.9] and evaluate the second component at $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}=\{z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}\}$, thus obtaining a map $-\times \breve {\partial }\breve {\diamond }^{\mathrm {lr}}\colon \mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })\to \mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$ (respectively, $-\times \breve {\partial }\breve {\diamond }^{\mathrm {lr}}\colon \mathrm {Hur}(\diamond,\partial )\to \mathrm {Hur}(\diamond,\partial )_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$).

Definition 4.13 We denote by $\hat {\mu }^{\diamond }_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$ the composition

We denote by $\check \sigma$ the composition of $\hat {\mu }^{\diamond }_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$ with the covering projection

\begin{gather*} p_{G,G^{\rm op}}\colon \mathrm{Hur}(\breve{\diamond}^{\mathrm{lr}},\breve{\partial})_{\breve{\partial}\breve{\diamond}^{\mathrm{lr}}}\to \mathrm{Hur}(\breve{\diamond}^{\mathrm{lr}},\breve{\partial})_{G,G^{\rm op}}\\ (\text{respectively, }p_{G,G^{\rm op}}\colon \mathrm{Hur}(\diamond,\partial)_{\breve{\partial}\breve{\diamond}^{\mathrm{lr}}}\to \mathrm{Hur}(\diamond,\partial)_{G,G^{\rm op}}). \end{gather*}

Here we regard $(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })$ (respectively, $(\diamond,\partial )$) as an lr-based nice couple, using the two points $z_{\diamond }^{\mathrm {l}}$ and $z_{\diamond }^{\mathrm {r}}$ of $\breve {\partial }\breve {\diamond }^{\mathrm {lr}}$. See Figure 5.

Figure 5. Left: the product $(t_1,\mathfrak {c}_1)(t_2,\mathfrak {c}_2)(t_3,\mathfrak {c}_3)$ of three configurations in $\breve {\mathrm {HM}}$; for a given $\underline {w}\in \mathring {\Delta }^3$ the barycentres $b,b_\varepsilon ^+,b_\varepsilon ^-$ are shown as dotted, vertical lines. We use the letter ‘$q$’ to represent the ${\mathcal {Q}}$-valued monodromies around points in $\mathring {{\mathcal {R}}}_{\mathbb {R}}$. Right: the image of $(\underline {w};\underline {t},\underline {\mathfrak {c}})$ along $\check \sigma$; the question marks suggest that, because of the quotient by the $G\times G^{\rm op}$-action, the monodromies of the loops spinning around $z_{\diamond }^{\mathrm {l}}$ and $z_{\diamond }^{\mathrm {r}}$ are just not defined.

Roughly speaking, $\hat {\mu }^{\diamond }_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$ improves the effect of $\hat {\mu }^\diamond$ by forcing the presence of $z_{\diamond }^{\mathrm {l}}$ and $z_{\diamond }^{\mathrm {r}}$ in support of the configuration $\hat {\mu }^\diamond (\underline {w};\underline {t},\underline {\mathfrak {c}})$, which is already supported in $\breve {\diamond }^{\mathrm {lr}}$ (in $\diamond$); if either point $z_{\diamond }^{\mathrm {l}},z_{\diamond }^{\mathrm {r}}$ is already in support of $\hat {\mu }^\diamond (\underline {w};\underline {t},\underline {\mathfrak {c}})$, then its local monodromy does not change when passing to $\hat {\mu }^{\diamond }_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}(\underline {w};\underline {t},\underline {\mathfrak {c}})$. The further composition $\check \sigma$ forgets the monodromy information around the two points $z_{\diamond }^{\mathrm {l}}$ and $z_{\diamond }^{\mathrm {r}}$ of the support of $\hat {\mu }^{\diamond }_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}(\underline {w};\underline {t},\underline {\mathfrak {c}})$.

The proof of Proposition 4.15 is in Appendix A.4. Since the quotient map $B\mathring {\mathrm {HM}}\to \bar {B}\mathring {\mathrm {HM}}$ (respectively, $B\breve {\mathrm {HM}}\to \bar {B}\breve {\mathrm {HM}}$) is a weak equivalence, Theorem 4.1 reduces to proving that the map $\bar \sigma$ is a weak equivalence. In fact, we will prove surjectivity of $\sigma _*$ and injectivity of $\bar \sigma _*$ on homotopy groups.

4.2 Surjectivity on homotopy groups

We fix $q\geq 0$ and want to show that $\sigma _*\colon \pi _q( B\mathring {\mathrm {HM}})\to \pi _q(\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{G,G^{\rm op}})$ (respectively, $\sigma _*\colon \pi _q(B\breve {\mathrm {HM}})\to \pi _q(\mathrm {Hur}(\diamond,\partial )_{G,G^{\rm op}})$) is surjective. For $q=0$ this will imply, in particular, that $\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{G,G^{\rm op}}$ (respectively, $\mathrm {Hur}(\diamond,\partial )_{G,G^{\rm op}}$) is connected.

Note that $\mathring {\mathrm {HM}}_\emptyset$ (respectively, $\breve {\mathrm {HM}}_\emptyset$) is a contractible topological monoid, hence the subspace $B\mathring {\mathrm {HM}}_\emptyset \subset B\mathring {\mathrm {HM}}$ (respectively, $B\breve {\mathrm {HM}}_\emptyset \subset B\breve {\mathrm {HM}}$) is also contractible. Moreover, the map $\sigma$ sends $B\mathring {\mathrm {HM}}_\emptyset$ (respectively, $B\breve {\mathrm {HM}}_\emptyset$) constantly to the basepoint $\mathfrak {c}^{\mathrm {lr}}$.

In order to prove that $\sigma$ induces an isomorphism on $q$-homotopy groups, it suffices to prove that the map $\sigma _*\colon \pi _q(B\mathring {\mathrm {HM}},B\mathring {\mathrm {HM}}_\emptyset )\to \pi _q(\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{G,G^{\rm op}})$ (respectively, $\sigma _*\colon \pi _q(B\breve {\mathrm {HM}},B\breve {\mathrm {HM}}_\emptyset )\to \pi _q(\mathrm {Hur}(\diamond,\partial )_{G,G^{\rm op}})$) is an isomorphism, i.e. we can consider relative homotopy groups.

To show that $\sigma _*$ is surjective, represent an element of $\pi _q(\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{G,G^{\rm op}})$ (of $\pi _q(\mathrm {Hur}(\diamond,\partial )_{G,G^{\rm op}})$) by a map

\[ f\colon D^q\to\mathrm{Hur}(\breve{\diamond}^{\mathrm{lr}},\breve{\partial})_{G,G^{\rm op}}\quad (\text{respectively, }f\colon D^q\to\mathrm{Hur}(\diamond,\partial)_{G,G^{\rm op}}) \]

sending $\partial D^q$ to the basepoint $\mathfrak {c}^{\mathrm {lr}}$. Thus, for all $v\in D^q$ we have an orbit $f(v)=[\mathfrak {c}_v]_{G,G^{\rm op}}$ of the action of $G\times G^{\rm op}$ on $\mathrm {Hur}(\breve {\diamond }^{\mathrm {lr}},\breve {\partial })_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$ (on $\mathrm {Hur}(\diamond,\partial )_{\breve {\partial }\breve {\diamond }^{\mathrm {lr}}}$). We choose for all $v\in D^q$ a representative $\mathfrak {c}_v=(P_v,\psi _v,\varphi _v)$ of $f(v)$; note that the sets $P_v\subset \breve {\diamond }^{\mathrm {lr}}$ (respectively, $P_v\subset \diamond$) do not depend on this choice; similarly the evaluation of $\psi _v$ and $\varphi _v$ is independent of the choice of $\mathfrak {c}_v$ on those elements of $\mathfrak {Q}(P_v)$ and $\mathfrak {G}(P_v)$ that can be represented by a loop contained in $[0,1]\times \mathbb {R}\,\setminus P_v$.

For all $v\in D^q$ we can find an open interval $J_v\subset (0,1)$ and a neighbourhood $v\in V_v\subseteq D^q$ such that for all $v'\in V_v$ the finite set $\Re (P_{v'})$ is disjoint from $J_v$. Using the compactness of $D^q$, we can then choose a cover of $D^q$ by finitely many open sets $V_i$ with corresponding open intervals $J_i\subset (0,1)$, satisfying $\Re (P_v)\cap J_i=\emptyset$ for all $v\in V_i$. After shrinking the intervals $J_i$ appropriately, we can assume they are disjoint. To fix notation, we assume that we have $r$ open sets $V_1,\ldots, V_r$, such that the corresponding intervals $J_1,\ldots,J_r$ appear in this order, from left to right, on $(0,1)$.

We choose numbers $A_i\in J_i$: note that $\mathbb {S}_{A_i,A_i}$ is disjoint from $P_v$ for all $v\in V_i$ and that the numbers $A_1,\ldots,A_r$ are all distinct. We fix weights $W_i\colon D^q\to [0,1]$ giving a partition of unity on $D^q$ subordinate to the covering $\{V_1,\ldots,V_r\}$. We also assume that for each $v\in D^q$ there are at least two distinct indices $1\le i\le r$ such that $W_i(v)>0$.