2.1 Fragmentation homotopy

Let $\{\mu _i\}_{i=1}^N$ be a partition of unity with respect to an open cover of M. We define a fragmentation homotopy with respect to this partition of unity. Let $\nu _0=0$ , and for $j>0$ , let $\nu _j$ be the function

$$\begin{align*}\nu_j(x)=\sum_{k=1}^j\mu_k(x). \end{align*}$$

We shall write $\Delta ^q$ for the standard q-simplex parametrized by

$$\begin{align*}\{ {\mathbf{t}}=(t_1,t_2,\dots,t_q); 0\leq t_1\leq \dots\leq t_q\leq 1\}.\end{align*}$$

We now consider the following map:

$$\begin{align*}H_1:M\times \Delta^q\to M\times \Delta^q, \end{align*}$$

$$\begin{align*}H_1(x, (t_1,t_2,\dots,t_q))=(x,(u_1,u_2,\dots,u_q)), \end{align*}$$

$$\begin{align*}u_i(x, {\mathbf{t}})=\nu_{\lfloor Nt_i \rfloor}(x)+\mu_{\lfloor Nt_i \rfloor+1}(x)(Nt_i-\lfloor Nt_i \rfloor). \end{align*}$$

Note that $u_i$ only depends on $t_i$ and x. Since $H_1({\mathbf {t}},x)$ preserves the x coordinate, we can define a straight line homotopy $H_t:M\times \Delta ^q\to M\times \Delta ^q$ from the identity to $H_1$ . As in Figure 1, the map $H_1$ is defined so that the gray area is mapped onto the union of the bold lines in the target where the union of bold lines is a subcomplex of $M\times \Delta ^q$ of dimension $n=\text {dim}(M)$ .

Figure 1 Fragmentation map for $N=3$ and $q=1$ . The bold lines are the images of $M\times \{0\}, M\times \{1/3\}, M\times \{2/3\}$ and $M\times \{1\}$ under the map $H_1$ .

It is easy to check that $H_t$ is compatible with the face maps $d_i: \Delta ^{q-1}\to \Delta ^q$ . Therefore, for any simplicial complex K, we still can define the homotopy $H_t: M\times K\to M\times K$ . Note that we can choose the integer N as large as we want, but, as we shall see in the proof of Theorem 2.1, we want to homotope a map $g:K\to \text {{Sect}}_c(\pi )$ , and the choice of N depends on the dimension of the parameter space K.

Note that the topological dimension of the subcomplex L is n. But if we choose any small open ball $B_{\epsilon }$ of radius (e.g., $2^{-q-1}\epsilon $ , where $q=\text {dim}(K)$ ), then the homotopical dimension of $L_{\epsilon }(K):= H_1((M\backslash B_{\epsilon })\times V(K))$ is $n-1$ . This is because the n-dimensional manifold $M\backslash B_{\epsilon }$ has homotopical dimension $n-1$ , meaning that it has the homotopy type of a CW complex of dimension $n-1$ .

The fragmentation map $H_1$ has the following useful property.

Now, we want to use this lemma to prove Theorem 2.1. To deform a family of sections of $\pi : E\to M$ , parametrized by a map $g:K\to \text {{Sect}}_c(\pi )$ , we consider its adjoint as a map $G: M\times K\to E$ . We also define the support of g over K with respect to the base section $s_0$ as follows.

We shall need the following lemma that uses the fiber of the map $\pi :E\to M$ is $(n-1)$ -connected to prove Theorem 2.1.

Proof. We think of the desired homotopy $G_t:M\times D^q\to E$ as a section of the pullback of $\pi : E\to M$ over $M\times D^q\times [0,1]$ . The map $G_0$ is the adjoint of g. Let $Z\subset M\times D^q$ be the subcomplex consisting of points $(x,t)$ so that $G_0(x,t)=s_0(x)$ . By the homotopy extension property, we will obtain the desired homotopy $G_t$ , if we show that $G_0$ can be extended to a section $\tilde {G}$ over

$$\begin{align*}M\times D^q\times\{0\}\cup Z\times [0,1]\cup L_{\epsilon}(D^q)\times [0,1],\end{align*}$$

so that $\tilde {G}$ on $M\times D^q\times \{0\}$ is the same as $G_0$ , on $Z\times [0,1]$ is given by $\tilde {G}(x,t,s)=s_0(x)$ and on $L_{\epsilon }(D^q)\times \{1\}$ is also given by $\tilde {G}(x,t,1)=s_0(x)$ . So far, we know how to define $\tilde {G}$ on $M\times D^q\times \{0\}\cup Z\times [0,1]$ . We extend it over $L_{\epsilon }(D^q)\times [0,1]$ with a prescribed value on $ L_{\epsilon }(D^q)\times \{1\}$ by obstruction theory. Note that the homotopical dimension of $ L_{\epsilon }(D^q)\times [0,1]$ is n, and the fiber of the pullback of $\pi $ over $M\times D^q\times [0,1]$ is $(n-1)$ -connected. Hence, all obstruction classes that live in $H^*( L_{\epsilon }(D^q)\times [0,1]; \pi _{*-1}(\text {fiber}))$ vanish, and we obtain the desired extension $\tilde {G}$ .

2.2 Proof of Theorem 2.1

The idea is roughly as follows. To deform a family $g\colon D^q\to \text {Sect}_c(\pi )$ to a family of sections in $\text {Sect}_{\epsilon }(\pi )$ , we use Lemma 2.7 to assume that for the family g, we have $G(L_{\epsilon }(D^q))=s_0(M)$ . We then use the fragmentation homotopy to deform this family so that for each $s\in D^q$ , the section $g(s)$ sends the ‘most’ part of M to $G(L_{\epsilon }(D^q))$ . For example in Figure 1, for each $s\in D^1$ , the support of the section $g(s)$ lies inside the support of one function from the partition of unity, which can be chosen to be very small.

More precisely, we shall prove that homotopy groups of the pair $(\text {Sect}_c(\pi ), \text {Sect}_{\epsilon }(\pi ))$ are trivial. To do so, we show that for any commutative diagram

(1)

there exists a homotopy of pairs $(g_t,f_t): (D^{q}, S^{q-1})\to (\text {Sect}_c(\pi ), \text {Sect}_{\epsilon }(\pi ))$ so that $f_0=f$ , $g_0=g$ and $g_1:D^q\to \text {Sect}_c(\pi )$ factors through $\text {Sect}_{\epsilon }(\pi )$ . We first use the condition in Section 6 to satisfy the following.

This is because there exists a fiberwise homotopy $h_t:E\to E$ that is the identity on $s_0(M)$ and whose time $1$ maps a neighborhood of $s_0(M)$ onto $s_0(M)$ . So we can define a homotopy $F_t(x,s)=h_t(F(x,s)), G_t(x,s)=h_t(G(x,s))$ where F and G are adjoints of f and g, respectively. These maps give a homotopy of the diagram 1, and it is easy to see that for every $s\in S^{q-1}$ , there exists a neighborhood $\sigma $ of s so that $\text {supp}({F_1}|_{\sigma })\subset \text {supp}(f)(s)$ . So from now on, we assume that f satisfies the claim.

To deform the family $g:D^q\to \text {Sect}_c(\pi )$ to a family in $\text {Sect}_{\epsilon }(\pi )$ , we choose a partition of unity $\{\mu _i\}$ for a neighborhood of $\text {supp}(g|_{D^q})$ so that each $\text {supp}(\mu _i)$ can be covered by a ball of radius $2^{-q-1}\epsilon $ . Let $H_t:M\times D^q\to M\times D^q$ be the fragmentation homotopy associated with this partition of unity. By Lemma 2.7, there exists a homotopy $G': M\times D^q\times [0,1/2]\to E$ so that $G^{\prime }_0$ is the adjoint of g. For all $s\in D^q$ and $t\in [0,1/2]$ , we have $\text {supp}(G^{\prime }_t(s))\subset \text {supp}(g(s)) $ , and at time $1/2$ , we have $G_{1/2}(L_{\epsilon }(D^q))=s_0(M)$ . Note that if $s\in S^{q-1}$ , then $G^{\prime }_t(s)$ lies in $\text {Sect}_{\epsilon }(\pi )$ . Therefore, $G^{\prime }_t$ gives a homotopy of the pairs $(D^{q}, S^{q-1})\to (\text {Sect}_c(\pi ), \text {Sect}_{\epsilon }(\pi ))$ .

Now, we use the fragmentation homotopy to define $G_t: M\times D^q\to E$

$$\begin{align*}G_t:= \begin{cases} G^{\prime}_t & 0\leq t\leq 1/2,\\ G^{\prime}_1\circ H_{2t-1} & 1/2 \leq t \leq 1. \end{cases} \end{align*}$$

To show that $G_t$ is the desired homotopy, we first need to show that $G_t(-,S^{q-1})$ is also in $\text {Sect}_{\epsilon }(\pi )$ for $1/2 \leq t \leq 1$ . Recall that by the claim, for every $x\in S^{q-1}$ , there exists a simplex $\sigma $ containing x so that $\text {supp}(f|_{\sigma })$ is contained in at most k balls of radius $2^{-k}\epsilon $ for some k. We showed that $\text {supp}(G^{\prime }_1|_{\sigma })$ also has the same property. Since the fragmentation homotopy preserves the M factor, $\text {supp}(G^{\prime }_1\circ H_{2t-1}|_{\sigma })$ also has the same property. Hence, $G_t(-,S^{q-1})$ lies in $\text {Sect}_{\epsilon }(\pi )$ . So $G_t$ induces a homotopy of the pair of the map $(g,f)$ .

Now, it is left to show that $G_1(-,s)$ lies in $\text {Sect}_{\epsilon }(\pi )$ for all $s\in D^q$ . Note that the section $G_1(-,s)$ is the same as the base section on $H_1^{-1}(L_{\epsilon }(D^q))\cap M\times \{s\}$ . Hence, by Lemma 2.5, the support of $G_1(-,s)$ can be covered by $q+1$ balls of radius $2^{-q-1}\epsilon $ . Therefore, $G_1(-,s)$ is $\text {Sect}_{\epsilon }(\pi )$ for all $s\in D^q$ .

Note that $\text {{Sect}}_{\epsilon }(\pi )$ , which is a subspace of $ \text {{Sect}}_c(\pi )$ , has a natural filtration whose filtration quotients are similar to the filtration quotients induced by the non-abelian Poincaré duality (see [Reference LurieLur, Theorem 5.5.6.6]).

We, in fact, show in Appendix 6 that this theorem implies the non-abelian Poincaré duality for the space of sections of $\pi : E\to M$ . To recall its statement, let $\text {Disj}(M)$ be the poset of the open subsets of M that are homeomorphic to a disjoint union of finitely many open disks. For an open set $U\in \text {Disj}(M)$ , let $\text {Sect}_c(U)$ denote the subspace of sections that are compactly supported, and their supports are covered by U. Although the non-abelian Poincaré duality holds for topological manifolds, to use the fragmentation idea, we assume that M admits a metric for which there exists $\epsilon>0$ , such that all balls of radius $\epsilon $ are geodesically convex. For example, this holds for all compact smooth manifolds.

Corollary 2.10 (Non-abelian Poincaré duality)

If the fiber of the map $\pi $ is $(n-1)$ -connected, the natural map

$$\begin{align*}\underset{U\in\text{Disj}(M)}{\textsf{hocolim }} \mathrm{{Sect}}_c(U)\to \mathrm{{Sect}}_c(\pi), \end{align*}$$

is a weak homotopy equivalence.

3.1 n-fold delooping via fragmentation

The key step in proving Theorem 3.4 is to show that if F has a fragmentation property, then the map

$$\begin{align*}F(D^n,\partial)\to F^f(D^n,\partial)\simeq \Omega^nF(D^n) \end{align*}$$

is a homology isomorphism. To do so, we filter $F(D^n)$ and $F^f(D^n)$ . Since $D^n$ is compact, the fragmentation property for F and $F^f$ implies that

$$\begin{align*}F_{\epsilon}(D^n)\xrightarrow{\simeq} F(D^n), \end{align*}$$

$$\begin{align*}F^f_{\epsilon}(D^n)\xrightarrow{\simeq} F^f(D^n). \end{align*}$$

The spaces $F_{\epsilon }(D^n)$ and $F^f_{\epsilon }(D^n)$ are naturally filtered by the number of balls that cover the supports. We shall denote these filtrations and the corresponding maps between them by

(3)

Note that the last vertical map is a weak equivalence because $F^f(D^n)$ is a section space of a bundle over contractible space $D^n$ with the fiber $F(D^n)$ . Therefore, the map j in the diagram 3 also is a weak homotopy equivalence.

Proposition 3.8. Let F be a good functor satisfying the hypothesis of Theorem 3.4. Now, if $j_1$ in the diagram 3 induces a homology isomorphism, so does the map

$$\begin{align*}F(D^n,\partial)\to F^f(D^n,\partial)\simeq \Omega^nF(D^n). \end{align*}$$

We first explain the strategy to prove that $j_1$ is a homology isomorphism before we embark on proving Proposition 3.8. We have the following general lemma about filtered spaces ([Reference MatherMat76, Lemma 2, Section 27]):

Lemma 3.9. Consider the commutative diagram of spaces

Suppose:

• • $X_{\infty }$ and $Y_{\infty }$ are the union of $X_i$ ’s and $Y_i$ ’s, respectively, and for each i, the pairs $(X_i, X_{i-1})$ and $(Y_i, Y_{i-1})$ are good pairs.Footnote ³

• • f, $\iota $ and $\iota '$ are weak homotopy equivalences.

• • The filtration is so that if $f_1$ is k-acyclic Footnote ⁴ for some k, then the induced map

  $$\begin{align*}\overline{f_N}:X_N/X_{N-1}\to Y_N/Y_{N-1} \end{align*}$$
  is $(2N+k-2)$ -acyclic for every integer $N>1$ .

Then, $f_1$ induces a homology isomorphism.

Proof. We can assume that the maps $f_i$ are inclusions by replacing them with the mapping cylinder of $f_i$ . Therefore, the filtration $(Y_p,X_p)$ of $(Y_{\infty },X_{\infty })$ gives rise to a spectral sequence whose first page is

$$\begin{align*}E^1_{p,q}=H_{p+q}(Y_p, Y_{p-1}\cup X_p).\end{align*}$$

It converges to the homology of the pair $(Y_{\infty },X_{\infty })$ , but this pair is weakly homotopy equivalent to the pair $(Y,X)$ . Since the first condition f is a weak homotopy equivalence, the spectral sequence converges to zero. Now, we suppose the contrary that $f_1$ is not a homology isomorphism, and we choose the smallest k so that $E^1_{1,k}=H_{k+1}(Y_1,X_1)\neq 0$ . Therefore, $f_1$ is k-acyclic and, by the third condition, $\overline {f_p}$ is $(2p+k-2)$ -acyclic, which implies that $E^1_{p,q}=H_{p+q}(Y_p, Y_{p-1}\cup X_p)=0$ for $q\leq p+k-2$ .

Hence, as is indicated in Figure 2, no nontrivial differentials can possibly hit $E^1_{1,k}$ , which contradicts the fact that the spectral sequence converges to zero in all degrees.

Figure 2 All the differentials that map to $E^1_{1,k}$ have trivial domains. We drew differentials on the first, second and third pages.

In order to apply Lemma 3.9 to the diagram 3, we need to establish the second condition of Lemma 3.9 for the diagram. The subtlety here is in the filtrations $F_k(-)$ and $F_k^f(-)$ , where we know that the support is covered by k small balls, but the data of these balls are not given. We shall define certain auxiliary spaces by adding the data of covering balls.

3.1.1 Semisimplicial resolutions

To study the filtration quotients in the diagram 3, we shall define auxiliary semisimplicial spaces.

For the definition of semisimplicial spaces and the relevant techniques, we follow [Reference Ebert and Randal-WilliamsERW19]. Briefly, what we need about semisimplicial spaces and their (fat) realizations are as follows. First, we need a semisimplicial map that is a weak homotopy equivalence in each degree that induces a weak homotopy equivalence between fat realizations ([Reference Ebert and Randal-WilliamsERW19, Theorem 2.2]). Second, there is a skeletal filtration on the fat realization that gives rise to a spectral sequence calculating the homology of the fat realization ([Reference Ebert and Randal-WilliamsERW19, Section 1.4]). Last, we need the technical lemma in [Reference Galatius and Randal-WilliamsGRW18, Proposition 2.8] that gives a useful criterion to prove that the augmentation map for an augmented semisimplicial space induces a weak homotopy equivalence after taking realizations.

Recall that we assumed that M is a geodesic space and a small positive $\epsilon $ exists so that all balls of radius $\epsilon $ are geodesically convex. We say that a subset U of M is $\epsilon $ -admissible if it is open, geodesically convex and it can be covered by an open ball of radius $\epsilon $ .

Definition 3.10. Let $\mathcal {O}_{\epsilon }(M)$ be the discrete poset of open subsets of M that can be covered by a union of k geodesically convex balls of radius at most $2^{-k}\epsilon $ for some positive integer k.

Definition 3.11. Let $CF_k(M)$ be the subspace of $F(M)^k$ consisting of k-tuples so that each one has support contained in one ball of radius $2^{-k}\epsilon $ . We define the subspace $DF_k(M)$ of $CF_k(M)$ to be degenerate k-tuples; that is, the union of their supports can be covered by $k_0$ balls of radius $2^{-k_0}\epsilon $ for some $k_0<k$ . We denote the quotient space $CF_k(M)/DF_k(M)$ by $NF_k(M)$ . Similarly, we can define $CF^f_k(M)$ , $DF^f_k(M)$ and $NF^f_k(M)$ .

The natural maps $NF_k(M)\to F_k(M)/F_{k-1}(M)$ and $NF^f_k(M)\to F^f_k(M)/F^f_{k-1}(M)$ are $(k!)$ -sheeted covers away from the base points. So if $\Sigma _k$ denotes the permutation group on k letters, we have the spectral sequence of the action whose $E^2$ -page is $H_p(\Sigma _k;H_q( NF_k(M)))$ converging to $H_{p+q}(F_k(M)/F_{k-1}(M))$ . Similarly, we have the same spectral sequence for $NF^f_k(M)$ , and the comparison of the spectral sequences implies the following.

Lemma 3.12. If the induced map $NF_k(M)\to NF^f_k(M)$ is j-acyclic, so is the map between the filtration quotients

$$\begin{align*}F_k(M)/F_{k-1}(M)\to F^f_k(M)/F^f_{k-1}(M). \end{align*}$$

Hence, to establish the third condition of Lemma 3.9 for the diagram 3, it is enough to study the acyclicity of the map $NF_k(M)\to NF^f_k(M)$ . To do so, we shall use the following semisimplicial spaces.

Definition 3.13. Let ${CF_k(M)}_{\bullet }$ be a semisimplicial space whose space of q-simplices is given by the tuples $(\sigma , (B_{ij}))$ , where $\sigma =(\sigma _1,\dots ,\sigma _k)\in CF_k(M)$ and $(B_{ij})$ is a $k\times (q+1)$ matrix of $(2^{-k}\epsilon )$ -admissible sets, such that $B_{ij}$ contains the support of $\sigma _i$ for all j (if the support of $\sigma _i$ is empty, then $B_{ij}$ ’s are just $(2^{-k}\epsilon )$ -admissible sets). We topologize the q-simplices as a subspace of $F(M)^k\times \mathcal {O}_{\epsilon }(M)^{kq+k}\kern-1pt$ .

Definition 3.14. We define sub-semisimplicial space $D{F_k(M)}_{\bullet }$ so that its q-simplices are given by pairs $(\sigma , (B_{ij}))$ so that for each $0\leq j\leq q$ , the closure of $\cup _i B_{ij}$ is covered by $k_0$ balls of radius $2^{-k_0}\epsilon $ for some $k_0<k$ .

We similarly define ${CF^f_k(M)}_{\bullet }$ and $D{F^f_k(M)}_{\bullet }$ .

Remark 3.15. If we keep track of the choice of $\epsilon $ in our notations, we have the useful identifications ${CF_k(M,\epsilon )}_{\bullet }={CF_1(M,2^{-k}\epsilon )}_{\bullet }^k$ and the same for $F^f\kern-1pt$ .

Lemma 3.16. The natural maps

$$\begin{align*}||{CF_k(M)}_{\bullet}||\to CF_k(M), \, ||D{F_k(M)}_{\bullet}||\to DF_k(M) \end{align*}$$

are all weak homotopy equivalencies where $||-||$ means the fat realization of a semisimplicial space (see [Reference Ebert and Randal-WilliamsERW19]). Similarly, the corresponding statement holds for $F^f\kern-1pt$ .

Proof. This is [Reference MatherMat76, Lemma in section 20] for the functor defined by Thurston. However, there is an oversight in that proof where Mather assumes that the augmentation map from the realization of semisimplicial sets to $F_k(M)$ is a fibration and says that it is enough to show that their fibers are contractible. To fix this oversight, we need Definition 3.3. The idea is to show that the augmentation maps are microfibrations with contractible fibers.

Let $\widetilde {CF_k(M, V_M)}$ be the subspace of $F(M)^k$ consisting of k-tuples, such that each one has a lax support (see Definition 3.2) in ball of radius $2^{-k}\epsilon $ . Similarly, we define $\widetilde {CF_k(M,V_M)}_{\bullet }$ . Hence, it is enough to show that

$$\begin{align*}\alpha\colon||\widetilde{CF_k(M,V_M)}_{\bullet}||\to \widetilde{CF_k(M, V_M)} \end{align*}$$

is a weak homotopy equivalence.

Let $S_{\bullet }$ be the simplicial set whose q-simplices are given by $q+1$ ordered $(2^{-k}\epsilon )$ -admissible sets. By the third condition of the goodness of the functor (see Definition 3.3), it is clear that $\widetilde {CF_k(M,V_M)}_{\bullet }\subset \widetilde {CF_k(M,V_M)}\times (S_{\bullet })^k$ is open. Similar to the proof of Lemma 6.4, this inclusion satisfies the conditions of [Reference Galatius and Randal-WilliamsGRW18, Proposition 2.8]. Therefore, the map $\alpha $ induced by the projection to the first factor is microfibration. To identify the fiber over $\sigma =(\sigma _1,\dots ,\sigma _k)$ , let $S_i$ be the set of $(2^{-k}\epsilon )$ -admissible sets containing the support of $\sigma _i$ . Let $S_{i\bullet }$ be the simplicial set whose q-simplices are given by mappings $[q]=\{0,1,\dots ,q\}$ to $S_i$ . Therefore, the realization of this simplicial set is contractible. The fiber over $\sigma $ can be identified with the fat realization of $S_{1\bullet }\times \cdots S_{k\bullet }$ . Since the fat realization and the realization for the simplicial sets are weakly equivalent, and the realization commutes with products ([Reference MilnorMil57]), we deduce that the fiber over $\sigma $ is contractible. The proof for the other augmentation map is similar.

Now, the strategy to check the third condition of Lemma 3.9 for the diagram 3 is as follows. We define a functor $\nu _N$ on spaces so that when we apply it to a k-acyclic map $f:X\to Y$ , we obtain a $(2N+k-2)$ -acyclic map $\nu _N(f):\nu _N(X)\to \nu _N(Y)$ . Then, we construct a homotopy commutative diagram

where the horizontal maps induce homology isomorphisms. In the next section, we shall define a suitable functor $\nu _k$ satisfying the desired properties.

3.1.2 A thick model of the suspension of a based space

To define the functor $\nu _k$ that receives a map from the above semisimplicial resolutions, we need to modify the definition of the suspension of a space. First, we define auxiliary simplicial sets associated with the manifold M with the fixed choice of $\epsilon $ .

Definition 3.17. Let $S(r)$ be the set of $(2^{-r}\epsilon )$ -admissible sets in M. Let $\Delta _{\bullet }(M,r)$ denote the simplicial set whose q-simplices are given by mappings $[q]$ into $S(r)$ (i.e., $(q+1)$ -tuple of elements in $S(r)$ ). Let $M_{\bullet }(r)$ be the subsimplicial set of $\Delta _{\bullet }(M,r)$ whose q-simplices consist of those admissible sets that the intersection of the entries of the tuple is nontrivial. Let $\partial M_{\bullet }(r)$ be the subsimplicial set of $\Delta _{\bullet }(M,r)$ whose q-simplices consist of those admissible sets that the intersection of the entries of the tuple and $\partial M$ is nontrivial.

Remark 3.18. For each k, the geometric realizations of $\Delta _{\bullet }(M,r)$ are contractible because it is a full simplex, and the geometric realizations of $M_{\bullet }(r)$ and $\partial M_{\bullet }(r)$ , by the nerve theorem, have the homotopy type of M and $\partial M$ , respectively.

Our modification of the suspension of a space X is as follows.

Definition 3.19. Let $\widetilde {\Sigma }^nX$ be the realization of the following semisimplicial space:

$$\begin{align*}\widetilde{\Sigma}_{\bullet}^nX(r):=\frac{(\Delta_{\bullet}(D^n,r)\times \{*\})\cup (D_{\bullet}^n(r)\times X)}{(t,x)\sim (t,x')\text{ if } t\in \partial D_{\bullet}^n(r)}. \end{align*}$$

Note that for each r, the space $\widetilde {\Sigma }^nX$ has the same homotopy type of the suspension $\Sigma ^nX$ , so we do not write the dependence on r. Because $D_{\bullet }^n(r)$ and $\partial D_{\bullet }^n(r)$ are semisimplicial sets that realize to the disk $D^n$ and the sphere $S^{n-1}$ , respectively. And $\Delta _{\bullet }(D^n,r)\times \{*\}$ is a contractible semisimplicial set that is glued to the base point. Note that we also have a natural projection $\pi :\widetilde {\Sigma }^nX(r)\to ||\Delta _{\bullet }(D^n,r)||$ .

Definition 3.20. Let $T_{k,\bullet }(M)$ be the subsimplicial set of $(\Delta _{\bullet }(M,k))^k$ whose q-simplices are given by matrices $(B_{ij})$ , $i=0,1,\dots ,q$ , $j=1,\dots ,k$ of admissible sets so that for each i, the union $\cup _j B_{ij}$ can be covered by $k_0$ open balls of radius $2^{-k_0}\epsilon $ for some $k_0<k$ . For $k=1$ , we define $T_{1,\bullet }=*$ .

Definition 3.21. We define $\theta _k(X)$ to be the pair

$$\begin{align*}((\widetilde{\Sigma}^nX)^k, (\pi^k)^{-1}(|T_{k,\bullet}(D^n)|)),\end{align*}$$

where $\pi ^k: (\widetilde {\Sigma }^nX)^k\to ||\Delta _{\bullet }(D^n, k)||^k$ is the natural projection. Let $\nu _k(X)$ denote the quotient

$$\begin{align*}(\widetilde{\Sigma}^nX)^k/(\pi^k)^{-1}(|T_{k,\bullet}(D^n)|). \end{align*}$$

Remark 3.22. Note that for $k=1$ , the space $\nu _1(X)$ has the homotopy type of $\Sigma ^nX$ .

We suppress n, the dimension from the notations $\theta _k(X)$ and $\nu _k(X)$ , as it is fixed throughout. The following technical lemma is the main property of the functor $\nu _k$ .

Lemma 3.23. If $f:X\to Y$ is j-acyclic, the induced map of pairs $\nu _k(f): \nu _k(X)\to \nu _k(Y)$ is $(j+n+2k-2)$ -acyclic.

Proof. Recall that the reduced suspension of X for a based space $(X,*)$ is the smash product $S^n\wedge X$ , and we represent points in this smash product by a pair $(s,x)$ , where $s\in S^n $ and $x\in X$ . First, it is not hard to see ([Reference MatherMat76, Section 24]) that the space $\nu _k(X)$ is homotopy equivalent to

$$\begin{align*}(S^n\wedge X)^k/\Delta_{\text{{fat}},k}(S^n,X), \end{align*}$$

where $\Delta _{\text {{fat}},k}(S^n,X)$ consists of tuples $\big ( (s_1,x_1), (s_2, x_2),\dots , (s_k, x_k)\big )$ , such that $s_i=s_j$ for some $i\neq j$ . We can further simplify the homotopy type of $\nu _k(X)$ by separating $S^n$ and X in the above quotient to obtain

$$\begin{align*}\nu_k(X)\simeq \big (S^{nk}/\Delta_{\text{{fat}},k}(S^n)\big )\wedge X^{\wedge k}\kern-1pt. \end{align*}$$

Note that if $f:X\to Y$ is j-acyclic, the long exact sequence for the homology of a pair implies that the induced map $f^{\wedge 2}: X\wedge X\to Y\wedge Y$ is $(j+1)$ -acyclic. Hence, one can inductively show that the induced map $f^{\wedge k}:X^{\wedge k}\to Y^{\wedge k}$ is $(j+k-1)$ -acyclic. Thus, it is enough to show that $\big (S^{nk}/\Delta _{\text {{fat}},k}(S^n)\big )$ is $(n+k-2)$ -acyclic. Using again the long exact sequence for the homology of a pair, we need to show that $\Delta _{\text {{fat}},k}(S^n)$ is $(n+k-3)$ -acyclic.

For $i\neq j$ , let $\Delta _{(i,j)}(S^n,k)\subset (S^n)^{\wedge k}$ be the subspace given by tuples $(s_1,s_2,\dots ,s_k)$ , where $s_i=s_j$ . Note that $\Delta _{(i,j)}(S^n,k)\simeq (S^n)^{\wedge (k-1)}$ . The fat diagonal $\Delta _{\text {{fat}},k}(S^n)$ is the union of $\Delta _{(i,j)}(S^n,k)\subset (S^n)^{\wedge k}$ for all pairs $(i,j)$ , where $i\neq j$ . These are not open subsets. Instead, they are sub-CW complexes, so we still can apply the Mayer-Vietoris spectral sequence for this cover to compute the homology of $\Delta _{\text {{fat}},k}(S^n)$ . Let $\Delta _{(i_1,j_1),\dots , (i_r,j_r)}(S^n,k)$ denote the intersection $\Delta _{(i_1,j_1)}(S^n,k)\cap \dots \cap \Delta _{(i_r,j_r)}(S^n,k)$ . Hence, we have

$$\begin{align*}E_{p,q}^1=\bigoplus_{(i_m,j_m)}H_q(\Delta_{(i_0,j_0),\dots, (i_p,j_p)}(S^n,k))\Longrightarrow H_{p+q}(\Delta_{\text{{fat}},k}(S^n)), \end{align*}$$

where the sum is over different tuples of pairs $(i_m,j_m)$ . Since the intersection $\Delta _{(i_0,j_0),\dots , (i_p,j_p)}(S^n,k)$ is a $n(k-p-1)$ -connective space, $E^1_{p,q}=0$ for $q<n(k-p-1)$ . Note that p is at most $k-2$ so we have $n+k-3< n(k-p-1)+p$ . However, if $p+q<n(k-1)-pn+p$ , we have $E^1_{p,q}=0$ , which implies that $\Delta _{\text {{fat}},k}(S^n)$ is $(n+k-3)$ -acyclic.

Now, we are ready to prove the third condition of Lemma 3.9 for the diagram 3.

3.1.3 Proof of Theorem 3.4 for $M=D^n$

We want to prove that the natural map

$$\begin{align*}F(D^n,\partial)\to F^f(D^n,\partial)\simeq \Omega^nF(D^n) \end{align*}$$

induces a homology isomorphism. To do this, we show that

Lemma 3.24. There exists a commutative diagram of pairs,

(4)

so that the horizontal maps are homology isomorphisms (by which we mean homology isomorphism on each member of the pair).

Before we prove this lemma, let us explain how the above lemma finishes the proof of Theorem 3.4 for $M=D^n$ . By Lemma 3.16, Lemma 3.23 and Lemma 3.24, the third condition of Lemma 3.9 for the diagram 3 holds. Hence, $j_1$ in the diagram 3 is a homology isomorphism. Recall that for $k=1$ , the pair $\theta _1(X)$ has the homotopy type of $(\Sigma ^nX,*)$ . Therefore, Proposition 3.8 follows from Lemma 3.24 for $k=1$ .

Construction 3.25. To define the horizontal map in the diagram 4, we first define a semisimplicial map

$$\begin{align*}f_{\bullet}:{CF_1(D^n, 2^{-k}\epsilon)}_{\bullet}\to \widetilde{\Sigma}_{\bullet}^nF(D^n,\partial)(k)= \frac{(\Delta_{\bullet}(D^n, k)\times\{*\})\cup(D_{\bullet}^n(k)\times F(D^n,\partial))}{(t,x)\sim (t,x')\text{ if } t\in \partial D_{\bullet}^n(k)}. \end{align*}$$

For a q-simplex $(\sigma , B_0,\dots ,B_q)$ on the left-hand side, we know that $\cap _i B_i$ contains $\text {supp}(\sigma )$ .

• • If $(B_0,\dots , B_q)$ is a q-simplex in $\partial D^n_{\bullet }(k)$ , then we send $(\sigma , B_0,\dots ,B_q)$ to the base point on the right-hand side.

• • If $(B_0,\dots , B_q)$ is a q-simplex in $D^n_{\bullet }(k)$ , but not in $\partial D^n_{\bullet }(k)$ , then the support of $\sigma $ lies inside $D^n$ . Therefore, $\sigma \in F(D^n,\partial )$ , so we send $(\sigma , B_0,\dots ,B_q)$ to the corresponding element in $ F(D^n,\partial )\times D_{\bullet }^n(k)$ .

• • And if $(B_0,\dots ,B_q)$ is in $\Delta _{\bullet }(D^n,k)$ , but not in $D_{\bullet }^n(k)$ , we send $(\sigma , B_0,\dots ,B_q)$ , to $(B_0,\dots ,B_q)$ in $\Delta _q(D^n,k)\times \{*\}$ .

Since the above map is a semisimplicial map, we could take the realization to obtain

(5) $$ \begin{align} f:||{CF_1(D^n, 2^{-k}\epsilon)}_{\bullet}||\to \widetilde{\Sigma}^nF(D^n,\partial). \end{align} $$

Recall from Remark 3.15 that ${CF_k(M,\epsilon )}_{\bullet }={CF_1(M,2^{-k}\epsilon )}^k$ . Therefore, the above construction gives rise to maps

$$\begin{align*}||{CF_k(D^n)}_{\bullet}||\to (\widetilde{\Sigma}^nF(D^n,\partial))^k. \end{align*}$$

Definition 3.20 and Definition 3.14 are so that the above map induces a map

$$\begin{align*}||D{F^f_k(D^n)}_{\bullet}||\to (\pi^k)^{-1}(||T_{k,\bullet}(D^n)||). \end{align*}$$

The lower horizontal map in diagram 4 is similarly defined.

Proof of Lemma 3.24

From the naturality of the construction, we obtain the commutative diagram 4. We show that the top horizontal map is a homology isomorphism. The proof for the bottom horizontal map is similar. We first show that the map f in 5 induces a homology isomorphism. Recall that for all semisimplicial spaces $X_{\bullet }$ , there is a spectral sequence $E^1_{p,q}(X_{\bullet })=H_q(X_p)$ that converges to $H_{p+q}(||X_{\bullet }||)$ . The map f induces a comparison map between spectral sequence

(6)

So to prove $f_*$ is an isomorphism, we need to show that $f_p$ induces a homology isomorphism. Note that we have the following commutative diagram:

(7)

where $\pi $ and $\tau $ are natural projections to the simplicial set $\Delta _{\bullet }(D^n,k)$ . Hence, to show that $f_p$ induces a homology isomorphism, it is enough to prove that $f_p$ induces a homology isomorphism on the fibers of $\tau $ and $\pi $ .

We have three cases:

• • If $\beta =(B_0,\dots , B_p)$ lies in $\Delta _p(D^n,k)$ , but not in $D^n_p(k)$ , then the fiber of $\tau $ consists of those elements in $F(D^n)$ that have empty support, which is a contractible space by Definition 3.3. The fiber of $\pi $ over $\beta $ is a point.

• • If $\beta =(B_0,\dots , B_p)$ lies in $D_p^n(k)$ , but not in $\partial D^n_p(k)$ , then the fiber of $\tau $ over $\beta $ is the subspace $F_c(\cap _jB_j)$ . The fiber of $\pi $ over $\beta $ is $F(D^n,\partial )$ . Note that by the second condition in Definition 3.3, the inclusion $F_c(\cap _jB_j)\hookrightarrow F(D^n,\partial )$ is a homology isomorphism.

• • If $\beta =(B_0,\dots , B_p)$ lies in $\partial D_p^n(k)$ , then the fiber of $\tau $ over $\beta $ is acyclic by the third condition of Definition 3.3, and the fiber of $\pi $ over $\beta $ is a point. Therefore, $f_p$ induces a homology isomorphism, which in turn implies that f induces a homology isomorphism.

Since ${CF_k(M,\epsilon )}_{\bullet }={CF_1(M,2^{-k}\epsilon )}^k$ . Therefore, the fact that f induces a homology isomorphism implies that the map

$$\begin{align*}||{CF_k(D^n)}_{\bullet}||\to (\widetilde{\Sigma}^nF(D^n,\partial))^k \end{align*}$$

is also a homology isomorphism. Similar to the diagram 7, one can fiber the map

$$\begin{align*}D{F^f_k(D^n)}_{\bullet}\to (\pi^k)^{-1}(T_{k,\bullet}(D^n)) \end{align*}$$

over $T_{k,\bullet }(D^n)$ to prove that

$$\begin{align*}||D{F^f_k(D^n)}_{\bullet}||\to (\pi^k)^{-1}(||T_{k,\bullet}(D^n)||) \end{align*}$$

is also a homology isomorphism.

Remark 3.26. Let U be an open subset of M that is homeomorphic to the disjoint union of Euclidean spaces of dimension n. The same proof as the case of $D^n$ implies that

$$\begin{align*}F_c(U)\to F^f_c(U)\simeq \prod_{\pi_0(U)}\Omega^nF(D^n,\partial), \end{align*}$$

is a homology isomorphism.

3.1.4 Proof of Theorem 3.4

Since both F and $F^f$ satisfy the fragmentation property, the spaces $F_c(M)\simeq F_{\epsilon }(M)$ and $F_c^f(M)\simeq F^f_{\epsilon }(M)$ can be filtered, and the natural map $F_{\epsilon }(M)\to F_{\epsilon }^f(M)$ respects the filtration. Hence, it is enough to show that the induced map between filtration quotients induces a homology isomorphism. Using Lemma 3.12, it is enough to prove that the induced map between pairs

$$\begin{align*}(F_k(M), DF_k(M))\to (F^f_k(M), DF^f_k(M)) \end{align*}$$

induces a homology isomorphism. Let us first show that $F_k(M)\to F^f_k(M)$ induces a homology isomorphism using the same idea as in the proof of Lemma 3.24. We use Lemma 3.16 to resolve $F_k(M)$ and $F^f_k(M)$ by $F_k(M)_{\bullet }$ and $F^f_k(M)_{\bullet }$ . Recall that $F_k(M, \epsilon )= (F_1(M, 2^{-k}\epsilon ))^k$ and $F^f_k(M, \epsilon )= (F^f_1(M, 2^{-k}\epsilon ))^k$ . Therefore, it is enough to show that

$$\begin{align*}F_1(M)_p\to F^f_1(M)_p \end{align*}$$

induces a homology isomorphism for each simplicial degree p. To do so, we consider the commutative diagram

(8)

where $\pi $ and $\tau $ are natural projections to the simplicial set $\Delta _{\bullet }(M)$ . Hence, to show that $f_p$ induces a homology isomorphism, it is enough to prove that $f_p$ induces a homology isomorphism on the fibers of $\tau $ and $\pi $ . Let $\sigma _p=(B_0, B_1, \dots , B_p)$ be a p-simplex in $\Delta _{p}(M)$ . There are two cases for the fibers of $\tau $ and $\pi $ over $\sigma _p$ :

• • The intersection of $B_i$ ’s is empty. Therefore, the preimages of $\sigma _p$ under $\tau $ and $\pi $ are contractible by the first condition in Definition 3.3.

• • The intersection of $B_i$ ’s is not empty. Given that the disks $B_i$ ’s are geodesically convex, their intersection is homeomorphic to a disk. Hence, the induced map on fibers over $\sigma _p$ is

  $$\begin{align*}F_c(\cap_i B_i)\to F^f_c(\cap_i B_i), \end{align*}$$
  which is a homology isomorphism, as we proved the main theorem for the disks relative to their boundaries in Section 3.1.3.

Similarly, we could see that $DF_k(M))\to DF^f_k(M)$ induces a homology isomorphism by fibering the semisimplicial resolutions $DF_1(M)_{\bullet }$ and $DF^f_1(M)_{\bullet }$ over the semisimplicial set $T_{1,\bullet }(M)$ . Hence, we conclude that the map between pairs

$$\begin{align*}(F_k(M), DF_k(M))\to (F^f_k(M), DF^f_k(M)) \end{align*}$$

induces a homology isomorphism.

Remark 3.27. One could give a different proof of Theorem 3.4 using Definition 3.3 and goodness of F directly. As in Appendix 6, we could use the notion of lax support to show that

$$\begin{align*}\underset{U\in{\mathbf{D}}(M)}{\textsf{hocolim }}F^f_c(U)\xrightarrow{\simeq} F^f_c(M). \end{align*}$$

Since F satisfies Definition 3.3, the same proof implies that

$$\begin{align*}\underset{U\in{\mathbf{D}}(M)}{\textsf{hocolim }}F_c(U)\xrightarrow{\simeq} F_c(M). \end{align*}$$

Using Remark 3.26 for $U\in {\mathbf {D}}(M)$ , we know that $F_c(U)\to F_c^f(U)$ is a homology isomorphism. Using the spectral sequence to compute the homology of the homotopy colimits and the bar construction model for the homotopy colimits, it is enough to prove that the natural map

$$\begin{align*}B_{\bullet}(F_c(-), {\mathbf{D}}(M), *)\to B_{\bullet}(F^f_c(-), {\mathbf{D}}(M), *) \end{align*}$$

is a homology isomorphism which easily follows from $F_c(U)\to F^f_c(U)$ being a homology isomorphism for all $U\in {\mathbf {D}}(M)$ .

4.1 Strategy to avoid the local data

Recall that the strategy is first to prove $F(D^n, \partial )\to \Omega ^n F^f(D^n)$ induces a homology isomorphism, and then, for a compact manifold M, the proof is exactly the same as Section 3.1.4. For smooth foliations without extra transverse structures, Thurston’s idea to avoid the local statement, as is explained in Mather’s note ([Reference MatherMat76]), is to consider disk model for $\overline {\mathrm {B}\Gamma }_n$ (see [Reference MatherMat76, Section 9]). More concretely, he proved that $\overline {\mathrm {BDiff}_c(\mathbb {R}^n)}\to \Omega ^n\overline {\mathrm {B}\Gamma }_n$ induces a homology isomorphism without using $\overline {\mathrm {BDiff}(\mathbb {R}^n)}\to \overline {\mathrm {B}\Gamma }_n$ as a homology isomorphism.

To recall the disk model for $\overline {\mathrm {B}\Gamma }_n$ , we define $F(D^n)$ to be the realization of the semisimplicial set whose q-simplices are given by germs of foliations on the total space of $\Delta ^q\times \mathbb {R}^n\to \Delta ^q$ around $\Delta ^q\times D^n$ that are transverse to the fibers. It is easy to show that $F(D^n)$ is homotopy equivalent to $\overline {\mathrm {B}\Gamma }_n$ . The advantage of the disk model is that one can define the support for the germ of the foliation by taking the intersection of the support of a representative with the disk $D^n$ , and it has Thurston’s fragmentation property. However, note that if a germ of a foliation is supported in an open set U in $\text {int}(D^n)$ , it would give a simplex in ${\text {Fol}}_c(U)\simeq \overline {\mathrm {BDiff}_c(U)}$ . In particular, we have $F(D^n, \partial )\simeq \overline {\mathrm {BDiff}_c(\mathbb {R}^n)}$ .

Similarly, we define $F^f(D^n)$ to be the space of maps $\text {Map}(D^n,\overline {\mathrm {B}\Gamma }_n)$ . Since $\overline {\mathrm {B}\Gamma }_n$ is at least $(n-1)$ -connected, it has the fragmentation property, and given that $D^n$ is contractible, we have $F(D^n)\simeq F^f(D^n)$ . Also, we have $F^f(D^n, \partial )\simeq \Omega ^n \overline {\mathrm {B}\Gamma }_n$ . Therefore, we have a diagram 3 for these choices, and Proposition 3.8 applies to prove that $\overline {\mathrm {BDiff}_c(\mathbb {R}^n)}\to \Omega ^n\overline {\mathrm {B}\Gamma }_n$ induces a homology isomorphism.

One can use the corresponding disk model for each case in Theorem 1.10 and follow the same strategy as Theorem 1.7. Hence, in each case, to show that the corresponding F satisfies the c-principle, we need to show that the fragmentation properties and the goodness conditions (Definition 3.3) for F are satisfied.

It is easy to see that these functors satisfy the first and the fourth conditions in Definition 3.3. Since the subspace of foliations with empty support is a point, it is therefore contractible. And the third and fourth conditions are obvious in this case. The second condition is also satisfied for these spaces of foliations because there exists a metric on the space of foliations that makes them complete metric spaces (see [Reference HirschHir73, Section 2]). Footnote ⁵ Hence, it is easy to see that the base point in these spaces, which is the horizontal foliation, makes them well-pointed. In particular, it is a strong neighborhood deformation retract. Therefore, similar to Lemma 6.3, all these functors satisfy Definition 3.3, meaning that enlarging the subspace of compactly supported foliations to lax compactly supported (which is an open subspace) does not change the homotopy type. Hence, to prove the goodness of these functors, we need to check the last two conditions in Definition 3.3.

The case of the contactomorphisms and PL homeomorphisms are similar, and given what we already know about the connectivity of the corresponding Haefliger spaces, as we shall see, we have all the ingredients to check the above conditions. Hence, we prove the c-principle for $\text {Fol}_c(M,\alpha )$ and $\text {Fol}_c^{\text {PL}}(M)$ first.

4.2 The case of the contactomorphisms and PL homeomorphisms

Haefliger’s argument in [Reference HaefligerHae71, Section 6] implies that $\overline {\mathrm {B}\Gamma _n^{\text {PL}}}$ is $(n-1)$ -connected. Haefliger showed ([Reference HaefligerHae70, Theorem 3]) that Phillips’ submersion theorem in the smooth category implies that $\overline {\mathrm {B}\Gamma _n}$ is n-connected. Given that Phillips’ submersion theorem also holds in the PL category ([Reference Haefliger and PoenaruHP64]), one could argue similar to the smooth case that $\overline {\mathrm {B}\Gamma _n^{\text {PL}}}$ is, in fact, n-connected. However, McDuff in [Reference McDuffMcD87, Proposition 7.4] also proved that $\overline {\mathrm {B}\Gamma _{2n+1,ct}}$ is $(2n+1)$ -connected, which is even one degree higher than what we need. Hence, $\text {Fol}^{f,\text {PL}}(D^n)$ is n-connected and $ \text {Fol}^f(D^{2n+1},\alpha )$ is $(2n+1)$ -connected. Therefore, the space of formal sections satisfies the fragmentation property. To prove the fragmentation property for $\text {Fol}^{\text {PL}}(D^n)$ and $ \text {Fol}(D^{2n+1},\alpha )$ , we shall use the following lemma.

The PL homeomorphism groups are known to be locally contractible ([Reference GauldGau76]) and, as is proved by Hudson ([Reference HudsonHud69, Theorem 6.2]), they also satisfy an appropriate isotopy extension theorem which gives the second condition in Lemma 4.1. Therefore, $\text {Fol}_c^{\text {PL}}(M)$ satisfies the fragmentation property. However, the group of contactomorphisms is also locally contractible ([Reference TsuboiTsu08, Section 3]), and it satisfies the second condition ([Reference RybickiRyb10, Lemma 5.2]). Hence, $\text {Fol}_c(M,\alpha )$ also satisfies the fragmentation property in the sense of Definition 1.5.

Now we are left to show that ${\text {Fol}}_c(-,\alpha )$ and ${\text {Fol}}_c^{\text {PL}}(-)$ are good functors in the sense of Definition 3.3.

Proof. Recall that we need to check the last two conditions in Definition 3.3. We focus on $\text {Fol}_c(-,\alpha )$ , and we mention where $\text {Fol}_c^{\text {PL}}(-)$ is different.

We may assume that U and V are balls $B(r)$ and $B(R)$ of radi $r<R$ in $\mathbb {R}^{2n+1}$ . We want to show that the induced map

$$\begin{align*}\iota\colon\overline{\mathrm{BCont}_c(B(r),\alpha_{st})}\to \overline{\mathrm{BCont}_c(B(R),\alpha_{st})} \end{align*}$$

is a homology isomorphism.

Note that for any topological group G, the homology of $\overline {\mathrm {B}G}$ can be computed by subchain complex $\text {S}_{\bullet }(\overline {\mathrm {B}G})$ of singular chains $\text {Sing}_{\bullet }(G)$ of the group G given by smooth chains $\Delta ^{\bullet }\to G$ that sends the first vertex to the identity (see section 1.4 of [Reference HallerHal98] for more detail). Given a chain c in $ \text {S}_{\bullet }(\overline {\mathrm {BCont}_c(B(R),\alpha _{st})})$ , to find a chain homotopy to a chain in $\overline {\mathrm {BCont}_c(B(r),\alpha _{st})}$ , we need an easy lemma ([Reference HallerHal98, Lemma 1.4.8]) which says that for every contactomorphism $h\in \mathrm {Cont}_{c,0}(B(R),\alpha _{st})$ that is isotopic to the identity, the conjugation by h induces a self-map of $ \overline {\mathrm {BCont}_c(B(R),\alpha _{st})}$ , which is the identity on homology. Hence, it is enough to show that there exists h a contactomorphism, isotopic to the identity, that shrinks the support of the given chain to lie inside $B(r)$ .

To find such compactly supported contraction, consider the following family of contactomorphisms $\rho _t: \mathbb {R}^{2n+1}\to \mathbb {R}^{2n+1} $

$$\begin{align*}\rho(x_0,x_1,\cdots, x_{2n+1})=(t^2.x_0,t.x_1,\cdots, t.x_{2n+1}). \end{align*}$$

For $t<1$ , it is a contracting contactomorphism, but it is not compactly supported. To cut it off, we use the fact that the family $\rho _t$ is generated by a vector field $\dot {\rho }_t$ .

Let $\lambda $ be a bump function that is positive on the support of the chain c and zero near the boundary of $B(R)$ . One wants to consider the flow of the vector field $\dot {\rho }_{\lambda .t}$ to cut off $\rho _t$ , but $\dot {\rho }_{\lambda .t}$ may not be a contact vector field. However, there is a retraction $\pi $ from the Lie algebra of smooth vector fields to contact vector fields on every contact manifold ([Reference BanyagaBan97, Section 1.4]). Briefly, the reason that this retraction exists is that there is an isomorphism (see [Reference BanyagaBan97, Proposition 1.3.11]) between contact vector fields on a contact manifold $(M, \alpha )$ and smooth functions by sending a contact vector field $\xi $ to $\iota _{\xi }(\alpha )$ , the contraction of $\alpha $ by $\xi $ . The retraction $\pi $ is defined by sending a smooth vector field to the contact vector field associated with the function $\iota _{\xi }(\alpha )$ . Therefore, the flow of $\pi (\dot {\rho }_{\lambda .t})$ gives a family of compactly supported contactomorphisms of $B(R)$ that shrinks the support of the chain c.

Hence, by conjugating by such contactomorphisms that are isotopic to the identity, we conclude that $\iota $ induces a homology isomorphism. The case of the PL foliations is much easier because the existence of such contracting PL homeomorphisms that are isotopic to the identity is obvious.

To check the last condition, we want to show that any chain c in $S_{\bullet }(\overline {\mathrm {BCont}(D^{2n+1}, \partial _1)})$ is chain homotopic to the identity. Recall that the chain complex $S_{\bullet }(\overline {\mathrm {BCont}(D^{2n+1}, \partial _1)})$ is generated by the set of smooth maps from $\Delta ^{\bullet }$ to $\mathrm {Cont}(D^{2n+1}, \partial _1)$ that send the first vertex to the identity contactomorphism. Note that this set is in bijection with the set of foliations on the total space of the projection $\Delta ^{\bullet }\times M\to \Delta ^{\bullet }$ that are transverse to the fibers M and whose holonomies lie in $\mathrm {Cont}(D^{2n+1}, \partial _1)$ . Thus, it is enough to show that each of these generators is chain homotopically trivial. Geometrically, this means that for each such foliation c on $\Delta ^{\bullet }\times M$ , there is a foliation on $\Delta ^{\bullet }\times [0,1]\times M$ transverse to the projection to the first two factors (i.e., it is a concordance) such that on $\Delta ^{\bullet }\times \{0\}\times M$ , it is given by c, and on $\Delta ^{\bullet }\times \{1\}\times M$ , it is the horizontal foliation. The idea is to ‘push’ the support of the foliation c towards the free boundary until the foliation becomes completely horizontal.

To do so, consider a small neighborhood U of $\partial _1$ in $D^{2n+1}$ that is in the complement of the support of the foliation c. Note that, as in the previous case, there is a contact contraction that maps $D^{2n+1}$ to U and is isotopic to the identity. Let us denote this contact isotopy by $h_t$ such that $h_0=\text {id}$ . Let $F\colon \Delta ^{\bullet }\times [0,1]\times D^{2n+1}\to \Delta ^{\bullet }\times D^{2n+1}$ be the map that sends $(s,t,x)$ to $(s, h_t(x))$ and $F_t$ be the map F at time t. Since $h_t$ is a contact isotopy, for each t, the pullback foliation $F_t^*(c)$ on $ \Delta ^{\bullet }\times D^{2n+1}$ also gives an element in $S_{\bullet }(\overline {\mathrm {BCont}(D^{2n+1}\partial _1)})$ . Therefore, the pullback foliation $F^*(c)$ on $\Delta ^{\bullet }\times [0,1]\times D^{2n+1}$ is a concordance from c to the horizontal foliation, which means that c is chain homotopic to the identity in the chain complex $S_{\bullet }(\overline {\mathrm {BCont}(D^{2n+1}, \partial _1)})$ .

Note that we only used that for each foliation c on $\Delta ^{\bullet }\times D^{2n+1}$ , there exists a neighborhood U away from the support of c and there is a contact embedding h that maps $D^{2n+1}$ into U which is also isotopic to the identity. Such embeddings isotopic to the identity also exist in the PL case. Therefore, $\text {Fol}_c(-,\alpha )$ and $\text {Fol}_c^{\text {PL}}(-)$ both satisfy the conditions of Definition 3.3.

As an application of this theorem, we could improve the connectivity of $\overline {\mathrm {B}\Gamma _{2n+1,ct}}$ .

Proof. We already know by McDuff’s theorem([Reference McDuffMcD87, Proposition 7.4]) that $\overline {\mathrm {B}\Gamma _{2n+1,ct}}$ is $(2n+1)$ -connected. To improve the connectivity by one, note that Rybicki proved ([Reference RybickiRyb10]) that $\overline {\mathrm {BCont}_c(\mathbb {R}^{2n+1})}$ has a perfect fundamental group. Therefore, its first homology vanishes. However, by Theorem 1.10, the space $\overline {\mathrm {BCont}_c(\mathbb {R}^{2n+1})}$ is homology isomorphic to $\Omega ^{2n+1}\overline {\mathrm {B}\Gamma _{2n+1,ct}}$ . Hence, we have

$$\begin{align*}0=H_1(\Omega^{2n+1}\overline{\mathrm{B}\Gamma_{2n+1,ct}};\mathbb{Z})=\pi_1(\Omega^{2n+1}\overline{\mathrm{B}\Gamma_{2n+1,ct}})=\pi_{2n+2}(\overline{\mathrm{B}\Gamma_{2n+1,ct}}), \end{align*}$$

which shows that $\overline {\mathrm {B}\Gamma _{2n+1,ct}}$ is $(2n+2)$ -connected.

However, as we mentioned in the introduction, the perfectness of the identity component of PL homeomorphisms $\text {PL}_0(M)^{\delta }$ of a PL manifold M is not known in general. Epstein ([Reference EpsteinEps70]) proved that $\text {PL}_c(\mathbb {R})^{\delta }$ and $\text {PL}_0(S^1)^{\delta }$ are perfect and left the case of higher dimensions as a question. In [Reference NarimanNar22, Theorem 1.4], the author used the c-principle of $\text {Fol}_c^{\text {PL}}(-)$ to prove the following.

5.1 Space of functions not having certain singularities

It would be interesting to see if the space of smooth functions on M not having certain singularities satisfies the fragmentation property. In particular, it would give a different proof of Vassiliev’s c-principle theorem ([Reference VassilievVas92, Section $3$ ]) using the fragmentation method. Let S be a closed semialgebraic subset of the jet space $J^r(\mathbb {R}^n;\mathbb {R})$ of codimension $n+2$ , which is invariant under the lift of $\mathrm {Diff}(\mathbb {R}^n)$ to the jet space. We denote the space of functions over M, avoiding the singularity set S by $F(M,S)$ . It is easy to check that F is a good functor satisfying the conditions in 3.3. To prove that F satisfies the c-principle, we need to check whether the functors F and $F^f$ satisfy the fragmentation property. It seems plausible to the author, as we shall explain, that using an appropriate transversality argument for stratified manifolds ought to prove fragmentation property for $F(M)$ . But, since we still do not know if fragmenting the space of functions $F(M)$ is independently interesting, we do not pursue it further in this paper.

Recall that to check that the space of formal sections $F^f$ has the fragmentation property, we have to show that $F(\mathbb {R}^n,S)$ is at least $(n-1)$ -connected. But, it is easy to see that $F(\mathbb {R}^n,S)$ is homotopy equivalent $J^r(\mathbb {R}^n,\mathbb {R})\backslash S$ (see [Reference KupersKup19, Lemma 5.13]), and this space by Thom’s jet transversality is at least even n-connected. Therefore, the space of formal sections $F^f$ has the fragmentation property.

It is still not clear to the author how to check whether F has the fragmentation property, but here is an idea inspired by the fragmentation property for foliations. We want to solve the following lifting problem up to homotopy:

where Q is a simplicial complex and P is a subcomplex. Let $\sigma \subset Q$ be a simplex. We can think of the restriction of g to each $\sigma $ by adjointness as a map $g\colon M\times \sigma \to \mathbb {R}$ . In Section 2.1, we defined the fragmentation homotopy $H_1\colon M\times \sigma \to M\times \sigma $ after fixing a partition of unity $\{\mu _i\}_{i=1}^N$ . We have the flexibility to choose this partition of unity. Note that for each point $\mathbf{t}\in \sigma $ , the space $H_1(M\times \{\mathbf{t}\})$ is diffeomorphic to M (see Figure 1). So the restriction of the map g to this space gives a smooth function on M. By jet transversality, we can choose the triangulation of Q fine enough so that for each simplex $\sigma $ and each point $\mathbf{t}\in \sigma $ , the restriction of g to $H_1(M\times \{\mathbf{t}\})$ avoids the singularity type S.

Let $f_0$ be a function in $F(M,S)$ that we fix as a base section to define the support of other functions with respect to $f_0$ . Similar to the proof of Theorem 2.1, consider the subcomplex $L(\sigma )$ , which is an n-dimensional subcomplex of $M\times \sigma $ , which is the union of finitely many manifolds $L_i$ that are canonically diffeomorphic to M. In fact, $L(\sigma )$ is a union of the graphs of finitely many functions $M\to \sigma $ inside $M\times \sigma $ . It is easy to choose the partition of unity so that $L(\sigma )$ is a stratified manifold. The goal is to find a homotopy of the family of functions in $F(M,S)$ denoted by $g_s\colon M\times \sigma \to \mathbb {R}$ so that $g_0=g$ , and $g_1$ restricted to $H_1(M\times \{\mathbf{t}\})$ for each $\mathbf{t}\in \sigma $ is in $F(M,S)$ , and most importantly, the restriction of $g_1$ to each $L_i$ is given by the base function $f_0$ . If we can find such homotopy, then the rest of the proof is similar to proving the fragmentation property for the space of sections in Theorem 2.1.

5.2 Foliation preserving diffeomorphism groups

Another interesting transverse structure is the foliation preserving case when we have a flag of foliations. To explain the functor in this case, let M be a smooth n-dimensional manifold and $\mathcal {F}$ be a codimension q foliation on M. Let $\text {Fol}_c(M, \mathcal {F})$ be the realization of the simplicial set whose k-simplices are given by the set of codimension $\text {dim}(M)$ foliations on $M \times \Delta ^k$ that are transverse to the fibers of the projection $M \times \Delta ^k\to \Delta ^k$ , and the holonomies are compactly supported diffeomorphisms of the fiber M that preserve the leaves of $\mathcal {F}$ .Footnote ⁶

To define the space of formal sections in this case, note that the foliation $\mathcal {F}$ on M gives a lifting of the classifying map for the tangent bundle of M to $\mathrm {B}\Gamma _q\times \mathrm {BGL}_{n-q}(\mathbb {R})$ , where $\Gamma _q$ is the Haefliger groupoid of germs of diffeomorphisms $\mathbb {R}^q$ . Now consider the diagram

(9)

where $\Gamma _{n,q}$ is the subgroupoid $\Gamma _n$ given by germs of diffeomorphisms of $\mathbb {R}^n$ that preserve the standard codimension q foliation on $\mathbb {R}^n$ (see [Reference Laudenbach and MeigniezLM16, Section 1.1] for more details). Let $\overline {\mathrm {B}\Gamma }_{n,q}$ denote the homotopy fiber of $\theta $ . Let ${\text {Fol}}^f_c(M, \mathcal {F})$ be the space of lifts of $\tau _{\mathcal {F}}$ to $\mathrm {B}\Gamma _{n,q}$ up to homotopy.Footnote ⁷ Since the trivial M-bundle $\text {Fol}_c(M, \mathcal {F})\times M$ is the universal trivial foliated M-bundle whose holonomy preserves the leaves of $\mathcal {F}$ , we have a homotopy commutative diagram

The adjoint of the top horizontal map induces the map $\text {Fol}_c(M, \mathcal {F}))\to \text {Fol}^f_c(M, \mathcal {F})$ . The method of this paper can be applied to show that $\text {Fol}_c(M, \mathcal {F})$ also satisfies the c-principle, but we pursue this direction and its consequences elsewhere.