Proof. The key point is that J is defined by maps that use only the first coordinate of the domain $\coprod _1^m \mathbb {R}^p$ and the first $n-p+1$ coordinates of the codomain $\mathbb {R}^n$ . Let $\omega \in \Omega \mathcal {L}_{m\cdot p}^n$ . Applying either of the two maps to $\omega $ gives an embedding $\coprod _1^{m-1} \mathbb {R}^{p+1} \ {\rightarrow }\ \mathbb {R}^{n+1}$ . We view $\omega $ as a family of embeddings $\omega _t:\coprod _1^m \mathbb {R}^p \times t \ {\rightarrow }\ \mathbb {R}^n \times t$ , $t\in [-1,1]$ , and hence, $\Omega J (\omega )$ as a family of embeddings $\coprod _1^{m-1} \mathbb {R}^p \times t \ {\rightarrow }\ \mathbb {R}^n \times t$ , $t\in [-1,1]$ . In these terms, $G \circ \Omega J (\omega ): \coprod _1^{m-1} \mathbb {R}^{p+1} \ {\rightarrow }\ \mathbb {R}^{n+1}$ is obtained from $\Omega J (\omega )$ by first taking the union of the latter family over both the domain and codomain and then extending by the standard link $(e_1,\dots ,e_{m-1})$ for $t_{p+1} \notin [-1,1]$ . Thus, we must essentially compare the union of the family $\Omega J(\omega )$ with the embedding $\coprod _1^{m-1} \mathbb {R}^p\times [-1,1] \ {\rightarrow }\ \mathbb {R}^n \times [-1,1]$ obtained by restricting the domain and codomain of $J \circ G(\omega )$ .

Let A, B and C be the subspaces in the definition of joining p-dimensional links in $\mathbb {R}^n$ . Then each of $\Omega J (\omega )$ and $J \circ G(\omega )$ is obtained by gluing embeddings of $A \times [-1,1]$ , $B \times [-1,1]$ , and $C \times [-1,1]$ . On $C \times [-1,1]$ , both are given by the restriction of the standard $(p+1)$ -dimensional $(m-1)$ -component long link, since the combined effect of the pre- and post-compositions by the affine-linear maps of the domain and codomain leaves the standard long link fixed. Recall that ${\varphi }_1$ is a diffeomorphism of $\mathbb {R}^{n-p+1}$ . On $A\times [-1,1]$ , $J \circ G(\omega )$ is given by

(2.2)

On each slice $A\times t$ , $\Omega J (\omega )$ is given by

(2.3)

Applying G to (2.3) means taking the union over $t\in [-1,1]$ , which has the effect of replacing $\omega _t$ by $G(\omega )$ and replacing each instance of by , thus yielding (2.2). On $B\times [-1,1]$ , we have a similar comparison of two expressions, except that the second factors are and , respectively.

Proof. For part (a), $\rho _S \circ \iota _S$ takes a long link f to . By assumption, $g \circ h:\mathbb {R}^2 \ {\rightarrow }\ \mathbb {R}^2$ is isotopic to , and an isotopy to produces a homotopy from $\rho _S \circ \iota _S$ to the identity map on $\mathcal {L}_{p_i:i\in S}$ . Similar homotopies yield the statements about maps in parts (b) and (c), from which the statements about the induced maps are immediate.

Proof. The left-hand side is an abelian group for all $i \geq 0$ because every $p_j\leq n-3$ . By Proposition 2.11 (a), for any $S\subset \{1,\dots , m\}$ , $\iota _S$ gives a splitting of the long exact sequence in homotopy of the fibration $\rho _S$ , and hence, $\pi _i \mathcal {L}_{p_1, \dots , p_m}^n \cong \ker (\rho _S)_* \oplus \mathrm {im} (\iota _S)_*$ . Repeatedly applying this decomposition for every $S \in \mathcal {S}$ shows that $\pi _i \mathcal {L}_{p_1, \dots , p_m}^n $ is the sum of the intersection and the sum on the right-hand side. One can show that this sum is direct by using induction on the cardinality of $\mathcal {S}$ and all three parts of Proposition 2.11.

Proof. The decomposition (2.7) follows from Corollary 2.13 with $\mathcal {S}=\{\{2\}\}$ . We obtain the decomposition (2.8) from Corollary 2.13 with $\mathcal {S}=\{\{1\},\{2\}\}$ , using the fact that $\mathrm {im} (\iota _1)_* \cap \mathrm {im} (\iota _2)_* =0$ . The latter fact holds because $(\rho _j\circ \iota _j)_*$ is the identity and $(\rho _j \circ \iota _k)_*=0$ if $j\neq k$ , by Proposition 2.11 (a) and (b), respectively.

Proof. We will specify a map homotopy inverse to p so that with $\partial =\pi \circ j \circ h$ , we get $G = \partial $ . We thus need to construct out of a loop of long links a path of pseudoisotopy embeddings in one dimension higher, which starts at the standard embedding e and ends at a long link. Write a map as

$$ \begin{align*} f=(f_1,f_2) : \mathbb{R} \times (\mathbb{R}^{p-1} \sqcup \mathbb{R}^{q-1}) &\ {\rightarrow}\ \mathbb{R}^{n-1} \\ f_1:(s,(t_1, \dots, t_p)) & \mapsto f_1(s)(t_1, \dots, t_{p-1})\\ f_2:(s,(t_1, \dots, t_q)) & \mapsto f_2(s)(t_1, \dots, t_{q-1}) \end{align*} $$

by identifying $\mathbb {R} \times (\mathbb {R}^{p-1} \sqcup \mathbb {R}^{q-1})$ with $(\mathbb {R} \times \mathbb {R}^{p-1}) \sqcup (\mathbb {R} \times \mathbb {R}^{q-1})$ . We define h by the formula

$$ \begin{align*} h(f) = (h(f)_1, \, h(f)_2) : [-1,1] \times (\mathbb{R}^p \sqcup \mathbb{R}^q) &\ {\rightarrow}\ \mathbb{R}^n \\ h(f)_1 : (s, (t_1, \dots, t_p)) & \mapsto \left( f_1(g(s,t_p)) (t_1, \dots, t_{p-1}), \, t_p \right) \\ h(f)_2 : (s, (t_1, \dots, t_q)) & \mapsto \left( f_2(g(s,t_q)) (t_1, \dots, t_{q-1}), \, t_q \right), \end{align*} $$

where $g(s,t)=(s+1)\rho (t)-1$ and $\rho :\mathbb {R}\ {\rightarrow }\ \mathbb {R}$ is a smooth cutoff function that is $-1$ for $t\leq -1/2$ and $1$ for $t \geq 1/2$ . We conclude the proof by noting that the function $g: \mathbb {R}^2\ {\rightarrow }\ \mathbb {R}$ satisfies the following properties:

• ○ g is a smooth function of t for all $s\in [-1,1]$ ,

• ○ $g(-1,t) \leq -1$ , so that $h(f)$ starts at e,

• ○ $g(s,t) \leq -1$ if $t \leq -1$ , so that at every time s, $h(f)$ is a pseudoisotopy embedding (standard on the bottom face of $I^p \sqcup I^q$ and all slices of $(I^{p-1} \sqcup I^{q-1}) \times \mathbb {R}$ below it),

• ○ $g(1,t) \geq 1$ if $t \geq 1$ , so that $h(f)$ ends at a long knot,

• ○ $g(s,t)$ is independent of t for all $t \geq 1$ , so that at every time s, $h(f)$ is a pseudoisotopy embedding (given by the same long knot on the top face of $I^p \sqcup I^q$ and all slices of $(I^{p-1} \sqcup I^{q-1}) \times \mathbb {R}$ above it), and

• ○ $g(s,1)=s$ , so that $\pi \circ j\circ h = G$ .

Proof. Consider the square

(4.2)

where the maps with domain are obtained by restricting an embedding to each of the two components. This square is $(2n-p-q-4)$ -cartesian. This fact is the improved version of Morlet’s disjunction lemma due to Goodwillie [Reference GoodwillieGoo90b, p. 6], who further generalized this result [Reference GoodwillieGoo90b, p. 12, Theorem D] [Reference Goodwillie and KleinGK15, Lemma 7.4]. One can see that the statement about the square (4.2) is equivalent to the improved version of Morlet’s disjunction lemma by considering the connectivity of the map between the fibers of the rows or columns. This property of the square means the map from to the homotopy limit of the rest of the square is $(2n-p-q-4)$ -connected. This homotopy limit consists of a point in ${\mathcal {P}}_p^n$ , a point in ${\mathcal {P}}_q^n$ , and a path $\Delta ^1 \ {\rightarrow }\ \ast $ joining their images; hence, it is homeomorphic to ${\mathcal {P}}_p^n \times {\mathcal {P}}_q^n$ . Another result of Goodwillie implies that the space ${\mathcal {P}}_p^n$ is $(2n-2p-5)$ -connected [Reference GoodwillieGoo90b, pp. 9-10] [Reference GoodwillieGoo90a, Lemma 3.16]. Thus, has connectivity at least $\min \{2n-p-q-5, \ 2n-2p-5, \ 2n-2q-5\} = 2n-2q-5$ .

Proof. Consider the 3-cube of spaces

This cube is $(3n-3p-6)$ -cartesian [Reference Goodwillie and KleinGK15, Lemma 7.4] (i.e., the map from ${\mathcal {P}}_{p,p,p}^n$ to the homotopy limit X of the rest of the cube is $(3n-3p-6)$ -connected). The space X consists of three points in , three paths $\Delta ^1 \ {\rightarrow }\ {\mathcal {P}}_p^n$ joining pairs of images of these points, and a map $\Delta ^2 \ {\rightarrow }\ \ast $ whose three faces are the images of the three paths. The last piece of data is superfluous, so X fibers over $({\mathcal {P}}_{p,p}^n)^3$ with fiber $(\Omega {\mathcal {P}}_p^n)^3$ . Since both ${\mathcal {P}}_p^n$ and ${\mathcal {P}}_{p,p}^n$ are $(2n-2p-5)$ -connected (by [Reference GoodwillieGoo90a, Lemma 3.16] and Lemma 4.2), X is $(2n-2p-6)$ -connected, hence so is ${\mathcal {P}}_{p,p,p}^n$ .

Proof. The maps of spaces underlying the sequence (4.3) are

(4.4)

We are interested in the maps they induce on $\pi _0$ . By Lemma 4.1, each such graphing map G fits into a fibration

(4.5)

where $0 \leq j \leq i+p-1$ . By Lemma 4.2, the connectivity c of the base space in this case satisfies

$$\begin{align*}c\geq (2(i+n-j)-2(i+q-j)-5) - j = 2n-2q-j-5 \geq 2n - i - p - 2q - 4. \end{align*}$$

If $i\leq 2n-p-2q-5$ , then $c\geq 1$ . Thus, each map in the sequence (4.4) is $1$ -connected and hence induces isomorphisms on $\pi _0$ . If $i=2n-p-2q-4$ , then $j=i+p-1$ yields $c\geq 0$ , while $j \leq i+p-2$ yields $c \geq 1$ . Hence in this case, the first map in the sequence (4.4) induces a surjection in $\pi _0$ , and the remaining maps induce isomorphisms on $\pi _0$ . Finally, if $p=q$ , then the first instance of $G_*$ is the graphing map from to , which by Theorem 3.2 is also injective.

Proof. By Corollary 2.13, the restriction fibration $\rho _2$ and the inclusion $\epsilon $ of its fiber yield a direct sum decomposition of $\pi _0$ of each space in the sequence (4.4), starting with

(4.9)

and ending with

(4.10)

Each graphing map $G_*$ preserves the direct sum decomposition and hence can be written as $G_*=G^{\prime }_*\oplus G^{\prime \prime }_*$ . We now apply Theorem 4.4. If $i \leq 2n-p-2q-5$ , each instance of $G^{\prime }_*$ and $G^{\prime \prime }_*$ is an isomorphism, so we can decompose as the first summand of (4.9) plus the second summand of (4.10). If $i=2n-p-2q-4$ , the last p instances of $G^{\prime \prime }_*$ are still isomorphisms because $p\geq 1$ , and all but the first instance of $G^{\prime }_*$ are isomorphisms. The first instance of $G^{\prime }_*$ is surjective, and by Theorem 3.2, it is also injective, so we obtain the direct-sum decomposition in this case too. The claim about the maps in (4.7) and (4.8) is immediate from this direct-sum decomposition.

The simplification of the group for $p=q$ holds because $\pi _i\mathcal {K}_p^n=0$ if $i\leq 2n-3p-4$ [Reference BudneyBud08, Proposition 3.9 (2)]. (This fact, together with the assumption on i, also explains the absence of a summand of $\pi _i\mathcal {K}_p^n$ in the decomposition (4.6)).

Proof. A result of Klein and Williams [Reference Klein and WilliamsKW07, Theorem 11.1], proven in further generality by Goodwillie and Klein [Reference Goodwillie and KleinGK15, Theorem E], states that the square

is $(2n - 2p - q - 3)$ -cartesian. We apply it to long links (or maps) of $P \sqcup Q = \mathbb {R}^p \sqcup \mathbb {R}^q$ in $N=\mathbb {R}^n$ , where $\mathrm {Map}(P,N) \simeq \ast $ . Proceeding as in the proof of Lemma 4.2, we get that for $i\leq 2n-2p-q-4$ ,

$$\begin{align*}\pi_i \mathrm{Emb}_c(\mathbb{R}^p, \, \mathbb{R}^n - \mathbb{R}^q) \cong \pi_i S^{n-q-1} \oplus \pi_i \mathcal{K}_p^n \cong \pi_i S^{n-q-1}, \end{align*}$$

where $\pi _i \mathcal {K}_p^n=0$ because $i \leq 2n-3p-4$ . An application of the splitting (2.7) yields the desired direct sum. The left-hand map in the square above is the map r from formula (3.4), of which $G^p$ is a section. This verifies the claim about the inclusion of the first summand.

Proof. We proceed by induction on m, with $m=\ell $ as the basis case. Suppose we know the theorem to be true for links with $m-1$ components. Consider the restriction fibration $\rho _S$ , $S=\{2,\dots ,m\}$ :

$$\begin{align*}\mathrm{Emb}_c\left(\mathbb{R}^{p_1}, \ \mathbb{R}^n - \coprod_{k=2}^m \mathbb{R}^{p_k} \right) \ {\rightarrow}\ \mathcal{L}_{p_1, \dots, p_m}^n \xrightarrow{\rho_S} \mathcal{L}_{p_2, \dots, p_m}^n. \end{align*}$$

By the direct-sum decomposition from Corollary 2.12 (with $\mathcal {S}=\{S\}$ ) and the induction hypothesis, it suffices to prove the analogue of the theorem where $\mathcal {L}^n_{p_1,\dots , p_m}$ is replaced by $\mathrm {Emb}_c\left (\mathbb {R}^{p_1}, \ \mathbb {R}^n - \coprod _{k=2}^m \mathbb {R}^{p_k} \right )$ .

Our strategy for proving the latter statement is to first separate knotting and braiding phenomena, and then translate braiding phenomena into Whitehead products on wedges of spheres. From Lemma 3.1, we have a retraction

$$ \begin{align*} r:\mathrm{Emb}_c\left(\mathbb{R}^{p_1}, \ \mathbb{R}^n - \coprod_{k=2}^m \mathbb{R}^{p_k} \right) \ {\rightarrow}\ \Omega^{p_1}\left(\mathbb{R}^{n-p_1} - \coprod_{k=2}^m \mathbb{R}^{p_k-p_1} \right). \end{align*} $$

Because r and its section induce a splitting of homotopy groups, it suffices to show that classes in the codomain and homotopy fiber of r are in $\mathrm {im} \left (\bigoplus _{|S|\leq \ell } (\iota _S)_*\right ) = \sum _{|S|\leq \ell } \mathrm {im} (\iota _S)_*$ .

Let F be the homotopy fiber of r, which consists of pairs $(f,\gamma )$ such that f is in the domain of r and $\gamma $ is a path from $r(f)$ to the standard embedding $e_{p_1}$ . Similarly, define $F'$ as the homotopy fiber of the map

$$\begin{align*}r':\mathrm{Emb}_c\left(\mathbb{R}^{p_1}, \ \mathbb{R}^n \right) \ {\rightarrow}\ \Omega^{p_1}\left(\mathbb{R}^{n-p_1} \right). \end{align*}$$

There is a homotopy equivalence $F\ {\rightarrow }\ F'$ induced by the inclusion $\mathbb {R}^n - \coprod _{k=2}^m \mathbb {R}^{p_k} \ {\rightarrow }\ \mathbb {R}^n$ and the affine-linear map that sends $(t_1^*,0,\dots ,0)$ to $0^n$ . Its inverse is induced by an affine-linear inclusion $\mathbb {R}^n \ {\rightarrow }\ \mathbb {R}^n - \coprod _{k=2}^m \mathbb {R}^{p_k}$ whose image lies in $(-\infty , t_2^*) \times \mathbb {R}^{n-1}$ . (Recall that $t_1^*, \dots , t_m^*$ are the first coordinates of the components $e_1,\dots , e_m$ of the standard long link.) Since the codomain of $r'$ is contractible, we have homotopy equivalences $\mathrm {Emb}_c\left (\mathbb {R}^{p_1}, \ \mathbb {R}^n \right ) \ {\rightarrow }\ F' \ {\rightarrow }\ F$ . In particular, any element in $\pi _*F$ comes from knots (i.e., is in the image of $\iota _1$ ).

The codomain of r is homotopy equivalent to $\Omega ^{p_1}\left (\bigvee _{k=2}^m S^{n-p_k-1}\right )$ . By the Hilton–Milnor theorem [Reference HiltonHil55] (which we may apply since $n-p_k\geq 3$ ) and the assumed upper bound on i, any element of $\pi _i$ of the latter space is in the image of the map

$$\begin{align*}\bigoplus_{S\subset \{2,\dots, m\}, |S|\leq \ell-1} \pi_{i+p_1} \left( \bigvee_{k \in S} S^{n-p_k-1}\right) \ {\rightarrow}\ \pi_{i+p_1} \left(\bigvee_{k=2}^m S^{n-p_k-1}\right); \end{align*}$$

indeed, if a Whitehead product of classes from $\ell $ different summands lies in $\pi _{i+p_1}$ , then $i +p_1\geq 1+ \sum _{k=m-\ell +1}^m (n-p_k-2)$ . A wedge-sum of $\ell -1$ spheres corresponds to a link with $\ell $ components, so the proof is complete.

Proof. We first check that closure induces a surjection on path components. Given a spherical embedding f, let $f_+$ be its restriction to $\coprod _1^m D^p$ , where $D^p$ is the upper hemisphere of $S^p$ . Find an isotopy $F_+=F_+(x,t)$ of $f_+$ to the embedding used to construct the closure of a long link (called $g=(g_1,\dots ,g_m)$ in Definition 2.5). By the isotopy extension theorem, $F_+$ extends to an isotopy F of spherical embeddings starting at f. Then $F(-,1)$ is in the image of the closure map and is isotopic to f, so $\widehat {\cdot }$ is surjective on $\pi _0$ .

For injectivity, we will use an intermediate space, which we will show has the same path components as $\mathcal {L}_{m \cdot p}^n$ . Fix a diffeomorphism ${\varphi }:\coprod _1^m D^p \ {\rightarrow }\ \left ( \coprod _{i=1}^m \{(t_i^*,0^{n-p-1})\} \times D^p\right )$ , where each summand of $D^p$ in the domain is the upper hemisphere of $S^p$ . We define the space $\mathrm {DbEmb}\left ( \coprod _1^m S^p, \, S^n \right )$ of disk-based embeddings as the space of pairs $(f,g)$ where $f \in \mathrm {Emb} \left ( \coprod _1^m S^p, \, S^n \right )$ and $g: D^n \ {\rightarrow }\ S^n$ is a smooth, orientation-preserving embedding such that

• ○ $g {\varphi }$ is the restriction of f to $\coprod _1^m D^p$ , and

• ○ $g(D^n) \cap \mathrm {im} f = f \left ( \coprod _1^m D^p_i \right )$ .

We topologize it as a subspace of $\mathrm {Emb}\left ( \coprod _1^m S^p, \, S^n \right ) \times \mathrm {Emb}(D^n, S^n)$ . This definition is motivated by the d-based links used by Habegger and Lin [Reference Habegger and LinHL90, Definition 2.1] for link maps with $p=1$ and $n=3$ .

We start by establishing a bijection on isotopy classes between disk-based embeddings and long links. The projection $(f,g)\mapsto g$ to the second factor is a fibration

(4.11) $$ \begin{align} \mathcal{L}_{m\cdot p}^n \ {\rightarrow}\ \mathrm{DbEmb}\left( \coprod_1^m S^p, \, S^n \right) \ {\rightarrow}\ \mathrm{Emb}^+(D^n, S^n), \end{align} $$

where $\mathrm {Emb}^+(D^n, S^n)$ is the space of orientation-preserving embeddings $D^n \ {\rightarrow }\ S^n$ . (The fact that this projection and (4.12) below satisfy the homotopy lifting property can be seen by lifting a homotopy to the space of embeddings of the n-manifold with boundary obtained from a neighborhood of $g(D^n) \cup f\left (\coprod _1^m S^p\right )$ .) The inclusion of the fiber over g is $f \mapsto (\widehat {f}, g)$ , where $\widehat {f}$ is the closure obtained via the images of $D^p$ under g. The base space is homotopy equivalent to $SO(n+1)$ by shrinking and linearizing. In particular, it is path-connected.

It now suffices to check that the boundary map $\pi _1SO(n+1) \ {\rightarrow }\ \pi _0 \mathcal {L}_{p\cdot m}^n$ in the long exact sequence of the fibration (4.11) is trivial. Find a neighborhood of $g(D^n)$ homeomorphic to $D^{n-1} \times I$ such that the induced inclusion $D^n \ {\rightarrow }\ D^{n-1} \times I$ is the standard one. The generator of $\pi _1SO(n+1)$ in this context can be represented by a loop of rotations of the $D^{n-1}$ factor through an angle of $2\pi $ . The effect on a long link f in the fiber is to send it to (the isotopy class of) the long link obtained by rotating the factor of $I^n$ in $I^n = I^{n-1} \times I$ by $2\pi $ , that is, to f itself. Thus, the action is trivial on $\pi _0$ , so the inclusion of the fiber in (4.11) is a bijection on $\pi _0$ .

We now connect to isotopy classes of spherical links. The projection $(f,g)\mapsto f$ of a disk-based embedding to the first factor is a fibration

(4.12) $$ \begin{align} \mathrm{DbEmb}\left( \coprod_1^m S^p, \, S^n \right) \ {\rightarrow}\ \mathrm{Emb}\left( \coprod_1^m S^p, \, S^n \right). \end{align} $$

We will show that it is injective on $\pi _0$ by showing that its fibers are path connected. Since the closure map factors through the map (4.12), this will establish injectivity on $\pi _0$ of the closure map.

Because $n-p \geq 3$ , $\pi _0$ of the base space in (4.12) is a group, so its path components are homotopy equivalent to each other. Therefore, it suffices to consider the fiber F over a standard trivial link. By considering a small neighborhood of the interval $[t_1^*, t_m^*] \times 0^{n-1} = \bigcup _{i=2}^m [t_{i-1}^*, t_{i}^*] \times 0^{n-1}$ in $D^n$ , we see that F is homotopy equivalent to the space of framed embeddings of $m-1$ intervals with interiors lying in $M:=D^n - \left ( \coprod _1^m S^p \right )$ and prescribed values and p-frames at the endpoints. In turn, F fibers over the space E of unframed such embeddings in M, with fiber $\widetilde {F}$ given by the space of such framings. By a linearization argument again, $\pi _0 \widetilde {F}$ is $\left (\pi _1(O(n-1), O(n-p-1)) \right )^m$ , which is trivial since $n-p\geq 3$ .

It remains to show that E is path-connected. Note that M has a handle decomposition with one n-disk and m handles of index $n-p-1$ . Let $A \in E$ . Since $n-p-1\geq 2$ , we can perform an isotopy of each sub-arc of A that lies in a handle, fixing its endpoints, so that it ends up in the n-disk. Since $n \geq 4$ , we can take each such isotopy to be an isotopy of A (i.e., we can avoid self-intersections). By a similar argument using that $n \geq 4$ , we can find a further isotopy to a fixed standard arc in the n-disk. Thus, E is path-connected. Hence, so is F, so the map (4.12) is injective on $\pi _0$ , and the closure map is bijective on $\pi _0$ .

Proof. We use the fibration

where $1 \leq j \leq 2n-3p-3$ . By Lemma 4.2, the pseudoisotopy embedding space is $(2n-2p-5)$ -connected. Thus,

is an isomorphism if $i \leq 2n-2p-5$ and surjective if $i=2n-2p-4$ . The claims about the sequence itself follow because $p\geq 2$ implies $2n-3p-3 \leq 2n-2p-5$ , and $p=1$ yields $2n-3p-3=2n-2p-4$ . By Lemma 4.9, . The proof of the statement for 2-component links is completed using the calculation of the latter group from Haefliger’s results [Reference HaefligerHae67, Théorème 10.7] [Reference HaefligerHae66, Corollary 8.14]. Each 3-fold direct sum comes from two summands of $\pi _0 \mathrm {Emb}(S^{2n-2p-3}, S^{3n-3p-3}) \cong \pi _0 \mathcal {K}_{2n-2p-3}^{3n-3p-3}$ and one summand of $\pi _{n-p-1}V_M(\mathbb {R}^{M+n-p-1})$ where $M\geq 2$ . This also proves the claim about $\mathrm {im} (\iota _j)_*$ .

In the setting of 3 components, we have an analogous fibration

$$\begin{align*}\Omega \mathcal{L}_{3\cdot(p+j-1)}^{n+j-1} \ {\rightarrow}\ \mathcal{L}_{3\cdot(p+j)}^{n+j} \ {\rightarrow}\ \mathcal{P}_{3\cdot(p+j)}^{n+j}. \end{align*}$$

By Lemma 4.3, the base space is $(2n-2p-6)$ -connected. Thus,

$$\begin{align*}\pi_{i} \mathcal{L}_{3\cdot(p+j-1)}^{n+j-1} \ {\rightarrow}\ \pi_{i-1} \mathcal{L}_{3\cdot(p+j)}^{n+j} \end{align*}$$

is an isomorphism if $i \leq 2n-2p-6$ and surjective if $i=2n-2p-5$ . If $p \geq 3$ , then $2n-3p-3\leq 2n-2p-6$ , while if $p=2$ , then $2n-3p-3=2n-2p-5$ . By Lemma 4.9, $\pi _0 \mathcal {L}_{3\cdot (2n-2p-3)}^{3n-3p-3} \cong \pi _0\mathrm {Emb}\left ( \coprod _1^3 S^{2n-2p-3}, \, S^{3n-3p-3} \right )$ . Finally, since $4(2n-2p-3)<3((3n-3p-3)-2)$ , we can apply another result of Haefliger [Reference HaefligerHae67, Théorème 9.4], which says that the latter group is given by the direct sum of isotopy classes of knots and links with fewer than 3 components together with one summand of $\pi _{3n-3p-3}S^{3n-3p-3}$ . The remaining 6 summands of $\mathbb {Z}$ or $\mathbb {Z}/2$ come from knots on each of the 3 components and links on each of the 3 pairs of components, and the 3 summands of $\pi _{2n-2p-3} S^{n-p-1}$ come from links on each of the 3 pairs of components. This yields the claimed groups in both parities as well as the claims about $\mathrm {im} (\iota _j)_*$ and $\bigcap _{|S|<3} \ker (\rho _S)_*$ .

Proof. This follows from a result of Fresse, Turchin and Willwacher [Reference Fresse, Turchin and WillwacherFTW], a special case of which was obtained by Songhafouo Tsopméné and Turchin [Reference Songhafouo Tsopméné and TurchinSTT18a] in higher codimensions, including all cases where $1=p\leq n-3$ . They describe these groups via a complex of graphs with leaves labeled by the link components. It is roughly dual to the graph complex described below in Section 5.1, except that there are no link strands, only labels by them, and the graphs are required to be connected. It generalizes work of Arone and Turchin [Reference Arone and TurchinAT15] from one component to multiple components. If all source components have dimension p and the target has dimension n, the degree of a graph is $(n-1)|E| - n |I| - p|L|$ , where $|E|$ , $|I|$ and $|L|$ are the numbers of edges, internal (i.e., non-leaf) vertices and leaves, respectively.

From this, one can deduce that only two types of graphs can contribute to $\pi _{2n-3p-3} \mathcal {L}_{p,p}^n$ . One type is the ‘tripod’ graph (the trivalent tree with 3 leaves) in degree $2n-3p-3$ , with the 4 possible leaf-labelings by $\{1,2\}$ , for $n-p$ odd. The other type is the ‘2-hair hedgehog’ graph (the trivalent graph with 2 leaves and one double edge) in degree $2n-2p-4$ , with the 3 possible leaf-labelings by $\{1,2\}$ , for $n-p$ even and $p=1$ . (Each of these graphs is 2-torsion in one parity of $n-p$ , due to certain orientation relations.) These are the first two graphs shown in [Reference Arone and TurchinAT15, Section 3.2, Table B]. (In addition, by [Reference Arone and TurchinAT15, Theorem 3.1], these are the only graphs that contribute to $\pi _i \mathcal {L}_{p,p}^n \otimes \mathbb {Q}$ , $i\leq 2n-3p-3$ , apart from a single edge with distinct leaf-labels in degree $n-2p-1$ , where we know $\pi _{n-2p-1}\mathcal {L}_{p,p}^n \cong \mathbb {Z}$ by Corollary 4.5.)

Proof. The definition of the coboundary operator ensures that with our conventions for graph orientations, these cochains are cocycles. They lie in a dimension which admits no nonzero coboundary map, so they represent nontrivial and moreover linearly independent cohomology classes.

Part (a) then follows from the fact that $H^{2n-6}(\mathcal {K}_1^n; \, \mathbb {Z}) \cong \mathbb {Z}$ , established in work of Turchin [Reference TurchinTur07].

For parts (b) and (d), we use the description of $H_*(\Omega \mathrm {Conf}(m, \mathbb {R}^{n-1}); \mathbb {R})$ , due to Cohen and Gitler [Reference Cohen and GitlerCG02, Theorem 2.3], as the space of horizontal chord diagrams modulo the 4T relations, where each chord has degree $n-3$ . (The cohomology is thus isomorphic to Vassiliev invariants of pure braids.) Write a horizontal chord between strand i and j as $t_{ij}$ , where $1\leq i < j \leq m$ , and write a horizontal chord diagram as a product of $t_{ij}$ . A basis of horizontal chord diagrams in dimension $d(n-3)$ (before the quotient by 4T) is given by all degree-d monomials in the $t_{ij}$ , and we visualize the dual basis of functionals by the same types of diagrams. The 4T relations are the homogeneous degree-2 elements $[t_{ij}, t_{jk} + t_{ik}]$ with $i,j,k$ distinct. Thus, $H^{2n-6}(\Omega \mathrm {Conf}(m, \mathbb {R}^{n-1}); \mathbb {R})$ is isomorphic to the subspace of functionals in $\mathrm {span}\{(t_{ij}t_{k\ell })^*\}_{i<j, k<\ell }$ which vanish on all elements $[t_{ij}, t_{jk} + t_{ik}]$ .

By a theorem on graphing braids in previous joint work [Reference Komendarczyk, Koytcheff and VolićKKV20, Theorem 5.19], the graphing map $G:\Omega \mathrm {Conf}(m, \mathbb {R}^{n-1}) \ {\rightarrow }\ \mathcal {L}_{m \cdot 1}^n$ induces a surjection $G^*$ in cohomology. That theorem also implies that for the link diagrams $\sum c_i \Gamma _i$ in parts (b) and (d), $G^* \circ I (\sum c_i \Gamma _i) \in H^{*}(\Omega \mathrm {Conf}(m, \mathbb {R}^{n-1}); \mathbb {R})$ corresponds to the result of remembering only the (duals of) horizontal chord diagrams. Hence, for $m=2$ , $\eta \mapsto (t_{12} t_{12})^*$ , and $(t_{12} t_{12})^*$ forms a basis for $H^{2n-6}(\Omega \mathrm {Conf}(2, \mathbb {R}^{n-1}); \mathbb {R})$ . The fact that $I(\eta )$ is the Hopf invariant is established in our joint work [Reference Komendarczyk, Koytcheff and VolićKKV, Example B.2]. Thus, part (b) is proven.

For $m=3$ , there are the diagrams $\mu , \nu _1, \nu _2, \nu _3, \rho _{12}^*\eta , \rho _{13}^*\eta $ and $\rho _{23}^*\eta $ , where $\rho _{ij}^*\eta $ is the result of putting $\eta $ on strands labeled i and j with no edges incident to the remaining third component (so $\rho _{ij}^*I(\eta )=I(\rho _{ij}^*\eta )$ ). We have $\mu \mapsto (t_{12}t_{13})^* + (t_{12}t_{23})^* + (t_{13}t_{23})^*$ , $\nu _i \mapsto (t_{ij}t_{ik})^* - (t_{ik}t_{ij})^*$ (where $i,j,k$ are distinct), and $\rho _{ij}^*\eta \mapsto (t_{ij}t_{ij})^*$ . By the theorem on graphing braids [Reference Komendarczyk, Koytcheff and VolićKKV20, Theorem 5.19], each of these must vanish on the 4T relations. One can directly check their linearly independence in $\mathrm {span}\{(t_{ij}t_{kl})^*\}_{i<j, k<\ell }$ . We now check the dimension of the space of horizontal 2-chord diagrams modulo 4T for $m=3$ . There are nine diagrams $t_{ij}t_{k\ell }$ , with $1\leq i<j\leq 3$ , $1\leq k<\ell \leq 3$ . The three 4T relations (each determined by the pair $i<j$ ) span a dimension-2 subspace, so the quotient by them has rank 7. Thus, for $m=3$ , our 7 cocycles form a basis for $H^{2n-6}(\Omega \mathrm {Conf}(m, \mathbb {R}^{n-1}); \mathbb {R})$ , and part (d) is proven. (The class $I(\mu )$ is an analogue of the triple linking number, while $\rho _{ij}^*I(\eta )$ is an analogue of the square the linking number between strands i and j, and $I(\nu _i)$ is an analogue of the product of the two different pairwise linking numbers involving strand i.)

Finally, for part (c), by Theorem 4.12 for $p=1$ , we have that has dimension 3 or 4 according to whether n is odd or even. Since ${\mathcal {L}}_{1,1}^n$ is a homotopy-associative H-space, the Milnor–Moore theorem [Reference Milnor and MooreMM65] says that its rational homology is the universal enveloping algebra on its rational homotopy. Let a denote a generator of $\pi _{n-3}{\mathcal {L}}_{1,1}^n \otimes \mathbb {Q}$ as well as its image in rational homology under the Hurewicz map. Under this map, the Whitehead product in homotopy corresponds, up to a sign, to the graded commutator $[ -,\, -]$ under the Pontryagin product $\cdot $ in homology. If n is even, then $[a,a]=\pm 2a\cdot a$ , while if n is odd, $[a,a]=0$ so $a \cdot a$ is not primitive. Thus, in either parity, the rational homology (or cohomology) in degree $2n-6$ is 4-dimensional. Hence, the linearly independent set in question is a basis.

Sketch of proof of Lemma 5.2

The proof is based on that of a result of Cattaneo, Cotta-Ramusino and Longoni [Reference Cattaneo, Cotta-Ramusino and LongoniCCRL02, Theorem 6.1]. First, on the space of configurations where fewer than 2 points lie in some neighborhood $N_i$ , any integral vanishes because on that space, the variation of F through the i-th copy of $S^{n-3}$ admits a homotopy back to the original singular link, thus yielding a degenerate family. This implies that if $\Gamma $ is not a chord diagram, then $\langle I(\Gamma ), F \rangle =0$ .

It remains to consider the integrals of chord diagrams on the space $\mathcal {C}$ of configurations where each $N_i$ contains exactly two points. The configuration points in a chord diagram can be identified by their order on the link components, so there is only one way for each $N_i$ to contain a pair of points (i.e., $\mathcal {C}$ is connected). If points j and k do not lie in the same $N_i$ , then ${\varphi }_{jk}$ restricted to $\mathcal {C}$ is nullhomotopic. Thus, for a chord diagram $\Gamma $ , $\langle I(\Gamma ), F\rangle \neq 0$ only if the chords of $\Gamma $ correspond exactly to the singularities resolved in F.

Finally, consider a factor ${\varphi }_{jk}^*\omega $ in $I(\Gamma )$ . Its integral over the configuration space $I \times I$ of two points on segments of the link in some $N_i$ produces an $(n-3)$ -form. The pairing of this form with the $(n-3)$ -parameter family coming from the resolution of the i-th double point is the same as the linking number of the sphere $S^{n-2}$ obtained from the $(n-3)$ -parameter family that varies one segment and (a closure of) the other segment. Then by construction, if the chords in $\Gamma $ correspond to the resolved singularities of F, $\langle I(\Gamma ), F\rangle $ is product of factors $\pm 1$ , and hence is itself $\pm 1$ .

This argument can be made more precise by taking the limit as the diameters of the neighborhoods $N_i$ and the spheres $S^{n-3}$ inside them approaches zero.

Proof. The overall strategy is to first prove all the statements for $p=1$ and then use diagram (5.11) to prove the statements for larger values of p. The work of Budney will imply that the bottom row, which involves only spaces of knots, consists almost entirely of isomorphisms; Theorems 3.2 and 4.11 give crucial information about the rows of graphing maps involving multiple components.

To prove the statements for $p=1$ , we will define families of 1-dimensional m-component long links, for $m=1,2,3$ , by resolving singular links and joining components. These families will be called F and $F'$ (for $m=3$ ), H, L, and $L'$ (for $m=2$ ), and K (for $m=1$ ). Via maps J that join components, we will have $(F\pm F') \mapsto (L\pm L') \mapsto K$ and $H\mapsto K$ by construction. Our first main task is to identify the homology classes of these four cycles using Lemmas 5.1 and 5.2, which includes showing that $F\pm F'$ is (up to a sign) the image of $[b_{21}, b_{31}]$ , H is (up to a sign) the image of the Hopf map, and K is a generator of $H_{2n-6}\mathcal {K}_1^n$ .

Families F and $F'$ of 3-component 1-dimensional pure braids: Take a singular 3-strand braid $f=(f_1,f_2,f_3): \coprod _1^3 \mathbb {R} \hookrightarrow \mathbb {R}^n$ with two double-points, given by $f_1(s_1)=f_2(t)$ and $f_1(s_2)=f_3(u)$ with $s_1<s_2$ . One can construct f by starting with a singular braid in $\mathbb {R}^3$ – for example, as in Figure 2 – and taking its image under an inclusion of coordinates $i:\mathbb {R}^3 \hookrightarrow \mathbb {R}^n$ . Then construct a $(2n-6)$ -parameter family F by resolving the singularities, as described before Lemma 5.2.

Figure 2 (a) The singular 3-strand braid $f=(f_1,f_2,f_3)$ used to build the family F. (b) The singular 3-strand braid $f'=(f_1^{\prime },f_2^{\prime },f_3^{\prime })$ used to build the family $F'$ .

Next, repeat this process, but with a singular 3-strand braid $f'=(f_1^{\prime }, f_2^{\prime }, f_3^{\prime })$ with two double-points, given by $f_1^{\prime }(s_1)=f_3^{\prime }(u)$ and $f_1^{\prime }(s_2)=f_2^{\prime }(t)$ with $s_1<s_2$ . Let $F'$ be the $(2n-6)$ -parameter family obtained by resolving the singularities of $f'$ . We orient it by ordering the parameters as $s_1, s_2, t, u$ and ordering the spheres of resolution in the order that the double-points are listed.

Both F and $F'$ can be taken to be families of pure braids, rather than arbitrary long links. As a result, we can use Lemma 5.1 (d). There, we noted that the graphing map $\Omega \mathrm {Conf}(3,\mathbb {R}^{n-1}) \ {\rightarrow }\ \mathcal {L}_{1,1,1}^{n}$ induces an injection in homology [Reference Komendarczyk, Koytcheff and VolićKKV20], and we gave a basis for the dual of its image. We apply Lemma 5.2 to the cycles F and $F'$ and to the seven graph cocycles $\gamma $ in that basis. Only the first graph in the expressions for $\mu $ and $\nu _1$ pairs nontrivially with F, and the same is true for the second graph in the expression for $\nu _1$ and $F'$ . Diagrammatically, F is dual to and $F'$ is dual to . That is, we have that

(5.9) $$ \begin{align} \begin{aligned} \langle I(\mu), F \rangle &=1 \\\langle I(\nu_1), F \rangle &= 1\\\langle I(\nu_i), F \rangle &= 0 \text{ for } i=2,3 \\\langle \rho_{ij}^*I(\eta), F \rangle &= 0 \text{ for } 1\leq i<j\leq 3 \end{aligned} \quad \begin{aligned} \langle I(\mu), F' \rangle &= 0 \\\langle I(\nu_1), F' \rangle &= \mp 1 \\ \langle I(\nu_i), F' \rangle &= 0 \text{ for } i=2,3\\ \langle \rho_{ij}^*I(\eta), F' \rangle &= 0 \text{ for } 1\leq i<j\leq 3, \end{aligned} \end{align} $$

where as before, the sign for n odd is shown above the sign for n even. The class $[b_{21}, b_{31}]$ is known to generate $\pi _{2n-6}\Omega \mathrm {Conf}(3,\mathbb {R}^{n-1})\otimes \mathbb {Q}$ . From our previous joint work [Reference Komendarczyk, Koytcheff and VolićKKV, Example 4.8], we have $\langle I(\mu ), [b_{21}, b_{31}] \rangle = 1$ (possibly up to a sign) and $\langle I(\gamma ), [b_{21}, b_{31}] \rangle = 0$ for all the other graph cocycles $\gamma $ appearing in (5.9). Hence, $F\pm F'$ (for n odd, respectively even) is homologous to $[b_{21}, b_{31}]$ , as desired.

Figure 3 The singular 2-component long link $\ell =(\ell _1, \ell _2)$ obtained by joining strand 2 to strand 3 in the singular braid $f=(f_1,f_2,f_3)$ . An isotopy takes this singular long link to a singular 2-strand pure braid.

Figure 4 The singular 2-component long link $\ell '=(\ell _1^{\prime }, \ell _2^{\prime })$ obtained by joining strand 2 to strand 3 in the singular braid $f'=(f^{\prime }_1,f^{\prime }_2,f^{\prime }_3)$ .

Figure 5 The singular 2-component braid $h=(h_1,h_2)$ used to construct the family H.

Families L and $L'$ of 2-component 1-dimensional long links by joining components: We next define $(2n-6)$ -dimensional families L and $L'$ of 2-component long links as the images of F and $F'$ under the map that joins the second and third components, as described just before the Theorem statement. We can also describe L and $L'$ as the resolutions of the singular links $\ell $ and $\ell '$ obtained by joining components of the singular links f and $f'$ . (That is, joining components commutes with resolving singularities.) These singular links are shown in Figures 3 and 4.

By Lemma 5.2,

$$ \begin{align*} \begin{aligned} \langle I(\eta), L \rangle &= 1 \\\langle I(\lambda), L \rangle &= 0 \\\langle \rho_i^*I(\kappa), L \rangle &= 0 \text{ for } i=1,2 \end{aligned} \quad \begin{aligned} \langle I(\eta), L' \rangle &= \left\{\begin{array}{ll} -1 & \text{ if } n \text{ is odd}\\0 &\text{ if } n \text{ is even}\end{array}\right. \\\langle I(\lambda), L' \rangle &= 1 \\\langle \rho_i^*I(\kappa), L' \rangle &= 0 \text{ for } i=1,2. \end{aligned} \end{align*} $$

Indeed, L pairs nontrivially with only the first diagram in formulas (5.2) and (5.4) for $\eta $ , and $L'$ pairs nontrivially with only the first diagram in formulas (5.3) and (5.5) for $\lambda $ . Diagrammatically, L is dual to and $L'$ is dual to .

A family H of 2-component 1-dimensional pure braids: Next, we construct a $(2n-6)$ -parameter family H of 2-component 1-dimensional long links. Start with an immersion $h=(h_1, h_2)$ which is an embedding except at two double-points, where the double-points are given by $h_1(s_1)=h_2(t_1)$ and $h_1(s_2)=h_2(t_2)$ where $s_1 < s_2$ and $t_1 < t_2$ . See Figure 5. Resolve the singularities to obtain a $(2n-6)$ -parameter family H out of h. We orient it by ordering the parameters as $s_1, s_2, t_1, t_2$ and ordering the resolution spheres in the order that the double-points are listed. By Lemma 5.2,

(5.10) $$ \begin{align} \begin{aligned} \langle I(\eta), H \rangle &=1 \\ \langle I(\lambda), H \rangle &= 0 \\ \langle \rho_i^*I(\kappa), H \rangle &= 0 \text{ for } i=1,2. \end{aligned} \end{align} $$

Diagrammatically, H is dual to and $L'$ is dual to . Thus, $L\pm L'$ and H represent linearly independent elements of , though L and H represent the same element.

We now identify H with the image of a better known element in the homotopy groups of spheres. Recall that if n is even, then the free part of $\pi _{2n-5}(S^{n-2})$ maps isomorphically onto $H_{2n-6}(\Omega S^{n-2})$ , which via graphing injects into . Lemma 5.1 (b) and the pairings in (5.10) show that if $n-2=2,4$ or $8$ , then H represents (up to a sign) the class of the pure braid coming from the Hopf fibration. More generally, this shows that $2H$ and represent the same class, since the Whitehead square of an even-dimensional identity map has Hopf invariant $\pm 2$ .

A family K of 1-dimensional long knots: Finally, we define a $(2n-6)$ -dimensional family K of long knots in $\mathbb {R}^n$ as the image of the family H of 2-component pure braids under the map described just before the Theorem statement. As noted earlier, joining commutes with resolving singularities. Thus, we can alternatively define K by applying an obvious extension of the map J to the singular link h to get a singular knot k, as shown in Figure 6(b), and then resolving the singularities of k. One could also obtain K without reference to multiple-component links, defining k directly as a singular knot prescribed by (or dual to) the first diagram in formula (5.1). Thus, K generates $\pi _{2n-6}\mathcal {K}_1^n$ because the latter definition matches the construction of a generator of $\pi _{2n-6}\mathcal {K}_1^n$ in Budney’s work [Reference BudneyBud08], where a quadrisecant counting argument is used to prove that it is a generator. (By Lemma 5.2, $\langle I(\kappa ), K \rangle = 1$ , but this implies only that K is nonzero in $H_{2n-6}(\mathcal {K}_1^n; \, \mathbb {Z}) \cong \pi _{2n-6}\mathcal {K}_1^n$ , so we need to use Budney’s result.)

Figure 6 The singular knot k obtained by joining the two components of the singular 2-component braid h.

Figure 7 The singular knot obtained by joining the two components of the singular link $\ell ' =(\ell _1^{\prime }, \ell _2^{\prime })$ .

Since $H \mapsto K$ and $2H$ was identified with , part (a) is proven in the case that $p=1$ . In particular, if the ambient dimension n is $4$ , $6$ or $10$ , graphing the pure braid associated to the Hopf fibration gives a generator of the first nontrivial homotopy group of the space of long knots in $\mathbb {R}^n$ .

We next prove part (b) for $p=1$ . We just have to consider the result of joining the two components in the cycle $L\pm L'$ . We saw that L is homologous to H and hence maps to K, so it remains to check that the result $K'$ of joining the components of $L'$ pairs trivially with $\kappa $ . Indeed, the singularities of $K'$ do not coincide with the chords of $\kappa $ , so by Lemma 5.1 (a) and Lemma 5.2, $K'$ is zero in homology, and part (b) is proven for $p=1$ . A picture of $K'$ is shown in Figure 7.