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Let \(\mathsf{BD}\) be a Breen–Deligne package with data \((n,f)\) and homotopy \(h\). Let \(\kappa , \kappa '\) be sequences of nonnegative real numbers. (In applications \(\kappa \) is a \((n,f)\)-suitable sequence.)

Then \(\kappa '\) is *adept* to \((\mathsf{BD}, \kappa )\) if for all \(i\) the pair \((\kappa _i / 2, \kappa '_{i+1} \kappa _{i+1})\) is \(h_i\)-suitable. (Recall that \(h_i\) is the homotopy map \(n_i \to n_{i+1}\).)

Let \(f\) be a basic universal map from exponent \(m\) to \(n\). We get an induced homomorphism of profinitely filtered pseudo-normed groups \(M^m \to M^n\) bounded by the maximum (over all \(i\)) of \(\sum _j |f_{ij}|\), where the \(f_{ij}\) are the coefficients of the \(n \times m\)-matrix representing \(f\).

This construction is functorial in \(f\).

Let \(f\) be a basic universal map from exponent \(m\) to \(n\), and let \((c_2, c_1)\) be \(f\)-suitable. We get an induced map

induced by the morphism of profinitely filtered pseudo-normed groups \(M^m \to M^n\).

This construction is functorial in \(f\).

A *Breen–Deligne data* is a chain complex in \(\textnormal{FreeMat}\).

Concretely, this means that it consists of a sequence of exponents \(n_0, n_1, n_2, \dots \in \mathbb N\), and universal maps \(f_i\) from exponent \(n_{i+1}\) to \(n_i\), such that for all \(i\) we have \(f_i \circ f_{i+1} = 0\).

A morphism of Breen–Deligne data is a morphism of chain complexes.

We will now construct an example of a Breen–Deligne package. In some sense, it is the “easiest” solution to the conditions posed above. The exponents will be \(n_i = 2^i\), and the homotopies \(h_i\) will be the identity. Under these constraints, we recursively construct the universal maps \(f_i\):

We leave it as exercise for the reader, to verify that with these definitions \((n, f, h)\) forms a Breen–Deligne package.

Let \(\mathsf{BD}\) be a Breen–Deligne package, \(N\) a power of \(2\), and let \(\kappa , \kappa '\) be sequences of nonnegative real numbers. Assume that \(\kappa '\) is adept to \((\mathsf{BD}, \kappa )\). Let \(h^N\) be the homotopy between \(\pi ^N_\mathsf{BD}\) and \(\sigma ^N_\mathsf{BD}\) defined in Def 1.1.11.

For all \(i\), the pair \((\kappa _i / N, \kappa '_{i+1} \kappa _{i+1})\) is \(h^N_i\)-suitable.

For every natural numbers \(N\), the endofunctor \(N \otimes \_ \) on \(\textnormal{FreeMat}\) induces an endofunctor of Breen–Deligne data.

Concretely, it maps a pair \((n, f)\) of Breen–Deligne data, to the pair \(N \otimes (n,f)\) consisting of exponents \(N \cdot n_i\) and universal maps \(N \otimes f_i\).

Let \(0 {\lt} r\) and \(0 {\lt} r' \le 1\) be real numbers, and let \(V\) be an \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-module. Let \(\mathsf{BD}= (n,f)\) be Breen–Deligne data, and let \(\kappa = (\kappa _0, \kappa _1, \kappa _2, \dots )\) be a sequence of constants in \(\mathbb R_{\ge 0}\) that is very suitable for \((\mathsf{BD}, r, r')\). Let \(M\) be a profinitely filtered pseudo-normed group with \(r'\)-action of \(T^{-1}\).

For every \(c \in \mathbb R_{\ge 0}\), the maps from Definition 1.5.9 induced by the universal maps \(f_i\) from the Breen–Deligne \(\mathsf{BD}= (n,f)\) assemble into a complex of normed abelian groups

Together, these complexes fit into a system of complexes with the natural restriction maps.

By Lemma 1.5.10 the differentials are norm-nonincreasing. It is clear that the restriction maps are also norm-nonincreasing, and therefore the system is admissible.

Let \(\mathsf{BD}= (n, f)\) be Breen–Deligne data, let \(r, r' \in \mathbb {R}_{\ge 0}\), and let \(\kappa = (\kappa _0, \kappa _1, \dots )\) be a sequence of nonnegative real numbers. We say that \(\kappa \) is \(\mathsf{BD}\)-*suitable* (resp. *very suitable* for \((\mathsf{BD}, r, r')\)), if for all \(i\), the pair \((\kappa _{i+1}, \kappa _i)\) is \(f_i\)-suitable (resp. *very suitable* for \((f_i, r, r')\)).

(Note! The order \((\kappa _{i+1}, \kappa _i)\) is contravariant compared to Definition 1.1.14. This is because of the contravariance of \(\hat V(\_ )\); see Definition 1.5.9.)

Similarly, for any \(c{\gt}0\), we have

the \(m\)-fold fibre product of \(\operatorname{Hom}(\Lambda ',M)_{\leq c}\) over \(\operatorname{Hom}(\Lambda ,M)_{\leq c}\).

Let \(\pi : X \to B\) be a surjective morphism of profinite sets, and let \(S_\bullet \to S_{-1}\), \(S_{-1} := B\), be its augmented Čech nerve. Let \(V\) be a semi-normed group. Then the complex

is exact. Furthermore, for all \(\epsilon {\gt} 0\) and \(f \in \ker (\widehat{V}(S_{m}) \to \widehat{V}(S_{m+1}))\), there exists some \(g\in \widehat{V}(S_{m-1})\) such that \(d(g) = f\) and \(‖g‖\leq (1+\epsilon ) \cdot ‖f‖\). In other words, the complex is normed exact in the sense of Definition 1.4.6.

A pseudo-normed group \(M\) is *CH-filtered* if each of the sets \(M_c\) is endowed with a topological space structure making it a compact Hausdorff space, such that following maps are all continuous:

the inclusion \(M_{c_1} \to M_{c_2}\) (for \(c_1 \le c_2\));

the negation \(M_c \to M_c\);

the addition \(M_{c_1} \times M_{c_2} \to M_{c_1 + c_2}\).

The pseudo-normed group \(M\) is *profinitely filtered* if moreover the filtration sets \(M_c\) are totally disconnected, making them profinite sets.

A *morphism* of CH-filtered pseudo-normed groups \(M \to N\) is a group homomorphism \(f \colon M \to N\) that is

*bounded*: there is a constant \(C\) such that \(x \in M_c\) implies \(f(x) \in N_{Cc}\);*continuous*: for one (or equivalently all) constants \(C\) as above, the induced map \(M_c \to N_{Cc}\) is a morphism of profinite sets, i.e. continuous.

The reason the two definitions of continuity are equivalent is that a continuous injection from a compact space to a Hausdorff space must be a topological embedding.

A morphism \(f \colon M \to N\) is *strict* if \(x \in M_c\) implies \(f(x) \in N_c\) (in other words, if we can take \(C = 1\) in the boundedness condition above).

Let \(r'\) be a positive real number. A CH-filtered pseudo-normed group \(M\) has an *\(r'\)-action of \(T^{-1}\)* if it comes endowed with a distinguished morphism of CH-filtered pseudo-normed groups \(T^{-1} \colon M \to M\) that is bounded by \(r'^{-1}\): if \(x \in M_c\) then \(T^{-1}x \in M_{c/r'}\).

A morphism of CH-filtered pseudo-normed groups with \(r'\)-action of \(T^{-1}\) is a morphism \(f \colon M \to N\) of CH-filtered pseudo-normed groups that commutes with the action of \(T^{-1}\).

Let \(V\) be a semi-normed group, and \(X\) a compact topological space. We denote by \(V(X)\) the normed abelian group of locally constant functions \(X \to V\) with respect to the sup norm. With \(\hat V(X)\) we denote the completion of \(V(X)\).

These constructions are functorial in bounded group homomorphisms \(V \to V'\) and contravariantly functorial in continuous maps \(f \colon X \to X'\).

Note in particular that \(V(f)\) and \(\hat V(f)\) are norm-nonincreasing morphisms of semi-normed groups.

There are two natural actions of \(T^{-1}\) on \(\hat V(M_{\le c})\). The first comes from the \(r'\)-action of \(T^{-1}\) on \(M\) which gives a continuous map

and thus a normed group morphism \(V(M_{\le c}) \to V(M_{\le cr'})\) which can be extended by completion to

The other comes from Lemma 1.5.2, using the \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-module \(V\). Again by extension to completion, we get a map

that we can compose with the map \(\hat V(M_{\le c}) \to \hat V(M_{\le cr'})\), obtained from the natural inclusion \(M_{\le cr'} \to M_{\le c}\). We thus end up with two maps

and we define \(\hat V(M_{\le c})^{T^{-1}}\) to be the equalizer of \((T^{-1})^*\) and \([T^{-1}]\). In other words, the kernel of \((T^{-1})^* - [T^{-1}]\).

Let \(d\) be the constant from Proposition 1.6.3. Let \(k {\gt} 1\) and \(c_0 {\gt} 0\) be real numbers such that

Let \(m\) be any natural number, and put

Finally, let \(c_0'\) be \(\frac{c_0}{r' \cdot n_i}\), where \(n_i\) is the \(i\)-th index in our fixed Breen–Deligne data.

Then \(i\)-th column in the double complex is \((k^2, K)\)-weak bounded exact in degrees \(\le m\) for \(c \ge c_0'\).

Let \(\Lambda \) be a polyhedral lattice, and \(S\) a finite set. Then for all positive integers \(N\) there is a constant \(d\) such that for all \(c{\gt}0\) one can write any \(x\in \operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq c}\) as

where all \(x_i\in \operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq c/N+d}\).

In other words, for all \(N\), there exists a \(d\) such that \(\operatorname{Hom}(\Lambda , \overline{\mathcal L}_{r'}(S))\) is \(N\)-splittable with error term \(d\).

Let \(\Lambda \) be a finite free abelian group, let \(N\) be a positive integer, and let \(\lambda _1,\ldots ,\lambda _m\in \Lambda \) be elements. Then there is a finite subset \(A\subset \Lambda ^\vee \) such that for all \(x\in \Lambda ^\vee =\operatorname{Hom}(\Lambda ,\mathbb Z)\) there is some \(x'\in A\) such that \(x-x'\in N\Lambda ^\vee \) and for all \(i=1,\ldots ,m\), the numbers \(x'(\lambda _i)\) and \((x-x')(\lambda _i)\) have the same sign, i.e. are both nonnegative or both nonpositive.

For any \(m\geq 1\), let \(\Lambda '^{(m)}\) be given by \(\Lambda '^m / \Lambda \otimes (\mathbb Z^m)_{\sum =0}\); for \(m=0\), we set \(\Lambda '^{(0)} = \Lambda \). Then \(\Lambda '^{(\bullet )}\) is a cosimplicial polyhedral lattice, the Čech conerve of \(\Lambda \to \Lambda '\).

In particular, \(\Lambda '^{(0)} = \Lambda \to \Lambda ' = \Lambda '^{(1)}\) is the diagonal embedding.

Let \(G\) and \(H\) be semi-normed groups, let \(K\) be a subgroup of \(H\) and \(C\) be a positive real number. A morphism \(f : G → H\) is \(C\)-surjective onto \(K\) if, for all \(x\) in \(K\), there exists some \(g\) in \(G\) such that \(f(g) = x\) and \(\| g\| ≤ C\| x\| \). If \(K = H\) we simply say \(f\) is \(C\)-surjective.

Let \(h\) be the homotopy packaged with \(\mathsf{BD}\), and let \(h^N\) denote the \(n\)-th iterated composition of \(h\) (see Def 1.1.11) which is a homotopy between \(\pi ^N\) and \(\sigma ^N \colon N \otimes \mathsf{BD}\to \mathsf{BD}\).

Let \(H \in \mathbb {R}_{\ge 0}\) be such that for \(i = 0, \dots , m\) the universal map \(h^N_i\) is bound by \(H\) (see Def 1.1.3).

Then the double complex satisfies the normed homotopy homotopy condition (Def 1.4.14) for \(m\), \(H\), and \(c_0\).

Let \(\Lambda ^{(\bullet )}\) be the cosimplicial polyhedral lattice of Definition 1.6.11, and recall from 1.6.12 that \(\operatorname{Hom}(\Lambda ^{(m)}, M)\) is a profinitely filtered pseudo-normed group with action of \(T^{-1}\).

Hence \(\operatorname{Hom}(\Lambda ^{(\bullet )}, M)\) is a simplicial profinitely filtered pseudo-normed group with action of \(T^{-1}\).

Now apply the construction of the system of complexes from Definition 1.5.11 to obtain a cosimplicial system of complexes

Now take the alternating face map cochain complex to obtain a system of double complexes, whose objects are

As final step, rescale the norm on the object in row \(m\) by \(m!\), so that all columns become admissible: the vertical differential from row \(m\) to row \(m+1\) is an alternating sum of \(m+1\) maps that are all norm-nonincreasing.

Let \(f\) be a universal map from exponent \(m\) to \(n\), and let \((c_2, c_1)\) be \(f\)-suitable. We get an induced map

that is the completion of \(V(f)\).

This construction is functorial in \(f\).

Let \(0 {\lt} r\) and \(0 {\lt} r' \le 1\) be real numbers. Let \(f\) be a universal map from exponent \(m\) to \(n\), and let \((c_2, c_1)\) be very suitable for \((f, r, r')\). Then

is norm-nonincreasing.

Let \(f = \sum _g n_g g\) be a universal map from exponent \(m\) to \(n\), and let \((c_2, c_1)\) be \(f\)-suitable. We get an induced map

that is the sum \(\sum n_g V(g)\).

This construction is functorial in \(f\).

Let \(\mathsf{BD}\) be a Breen–Deligne package, and let \(r, r'\) be nonnegative reals, such that \(0 {\lt} r {\lt} 1\) and \(0 {\lt} r' \le 1\). Let \(\kappa \) be any sequence of positive reals.

There exists a sequence \(\kappa '\) of nonnegative real numbers that is adept to \((\mathsf{BD}, \kappa )\).

Let \(x_0, x_1, \dots \) be a sequence of reals, and assume that \(\sum _{i=0}^\infty x_i\) converges absolutely. For every natural number \(N {\gt} 0\), there exists a partition \(\mathbb N = A_1 \sqcup A_2 \sqcup \dots \sqcup A_N\) such that for each \(j = 1,\dots ,N\) we have \(\sum _{i \in A_j} x_i \le (\sum _{i=0}^\infty x_i)/N + 1\)

Fix radii \(1{\gt}r'{\gt}r{\gt}0\). For any \(m\) there is some \(k\) such that for all polyhedral lattices \(\Lambda \) there is a constant \(c_0(\Lambda ){\gt}0\) such that for all profinite sets \(S\) and all \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-modules \(V\), the system of complexes

is \(\leq k\)-exact in degrees \(\leq m\) for \(c\geq c_0(\Lambda )\).

Let \(\Lambda \) be a finite free abelian group, and let \(\lambda _1, \ldots , \lambda _m \in \Lambda \) be elements. Let \(M \subset \operatorname{Hom}(\Lambda , \mathbb Z)\) be the submonoid \(\{ x \mid x(\lambda _i) \ge 0 \text{ for all $i = 1, \dots , m$}\} \). Then \(M\) is finitely generated as monoid.

Let \(\mathsf{BD}= (n,f,h)\) be a Breen–Deligne package, and let \(\kappa = (\kappa _0, \kappa _1, \kappa _2, \dots )\) be a sequence of constants in \(\mathbb R_{\ge 0}\) that is \(\mathsf{BD}\)-suitable. Fix radii \(1{\gt}r'{\gt}r{\gt}0\). For any \(m\) there is some \(k\) and \(c_0\) such that for all profinite sets \(S\) and all \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-modules \(V\), the system of complexes

is \(\leq k\)-exact in degrees \(\leq m\) for \(c\geq c_0\).

In other words, we are considering the following two categories:

the category whose objects are natural numbers, and whose morphisms are matrices;

the category with the same objects, but with Hom-sets replaced by the free abelian groups generated by the sets of matrices. We denote this latter category \(\textnormal{FreeMat}\).

Let \(\Lambda \) be a polyhedral lattice, and \(M\) a profinitely filtered pseudo-normed group.

Endow \(\operatorname{Hom}(\Lambda , M)\) with the subspaces

As \(\Lambda \) is polyhedral, it is enough to check the given condition on \(f\) for a finite collection of \(x\) that generate the norm.

These subspaces are profinite subspaces of \(M^\Lambda \), and thus they make \(\operatorname{Hom}(\Lambda , M)\) ito a profinitely filtered pseudo-normed group.

If \(M\) has an action of \(T^{-1}\), then so does \(\operatorname{Hom}(\Lambda , M)\).

Let \(C_\bullet ^\bullet \) be a system of complexes. For an integer \(m\geq 0\) and reals \(k \ge 1\), \(K \ge 0\) and \(c_0 \ge 0\), we say the datum \(C_\bullet ^\bullet \) is *\(\leq k\)-exact in degrees \(\leq m\) and for \(c\geq c_0\) with bound \(K\)* if the following condition is satisfied. For all \(c\geq c_0\) and all \(x\in C_{kc}^i\) with \(i\leq m\) there is some \(y\in C_c^{i-1}\) such that

Let \(C_\bullet ^\bullet \) be a system of complexes. If \(C_\bullet ^\bullet \) is weakly \(\leq k\)-exact in degrees \(\leq m\) and for \(c\geq c_0\) with bound \(K\) and if, for all \(c\geq c_0\) and all \(x\in C_{kc}^i\) with \(i\leq m\) such that \(dx = 0\) there is some \(y\in C_c^{i-1}\) such that \(x_{|c} = dy\) then, for every positive \(δ\), \(C_\bullet ^\bullet \) is \(\leq k\)-exact in degrees \(\leq m\) and for \(c\geq c_0\) with bound \(K + δ\).

Let \(C_\bullet ^\bullet \) be a system of complexes. For an integer \(m\geq 0\) and reals \(k \ge 1\), \(K \ge 0\) and \(c_0 \ge 0\), the datum \((C_c^\bullet )_c\) is *weakly \(\leq k\)-exact in degrees \(\leq m\) and for \(c\geq c_0\) with bound \(K\)* if the following condition is satisfied. For all \(c\geq c_0\), all \(x\in C_{kc}^i\) with \(i\leq m\) and any \(ε {\gt} 0\) there is some \(y\in C_c^{i-1}\) such that

Let \(r' {\gt} 0\) be a real number, and let \(S\) be a finite set. Denote by \(\overline{\mathcal L}_{r'}(S)\) the set

Note that \(\overline{\mathcal L}_{r'}(S)\) is naturally a pseudo-normed group with filtration given by

Let \(\Lambda \) be a polyhedral lattice, and \(S\) a profinite set. Then for all positive integers \(N\) there is a constant \(d\) such that for all \(c{\gt}0\) one can write any \(x\in \operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq c}\) as

where all \(x_i\in \operatorname{Hom}(\Lambda ,\overline{\mathcal L}_{r'}(S))_{\leq c/N+d}\).

In other words, for all \(N\), there exists a \(d\) such that \(\operatorname{Hom}(\Lambda , \overline{\mathcal L}_{r'}(S))\) is \(N\)-splittable with error term \(d\).

Let \(r' {\gt} 0\) and \(c \ge 0\) be real numbers, and let \(S\) be a finite set. The space \(\overline{\mathcal L}_{r'}(S)_{\le c}\) is the profinite limit of the finite sets

This endows \(\overline{\mathcal L}_{r'}(S)_{\le c}\) with the profinite topology. In particular, it is a profinitely filtered pseudo-normed group.

The natural action of \(T^{-1}\) on \(\overline{\mathcal L}_{r'}(S)\) restricts to continuous maps

In particular, \(\overline{\mathcal L}_{r'}(S)\) has an \(r'\)-action of \(T^{-1}\).

Let \(G\) and \(H\) be normed groups. Let \(K\) be a subgroup of \(H\) and \(f\) a morphism from \(G\) to \(H\). Assume that \(G\) is complete and \(f\) is \(C\)-surjective onto \(K\). Then \(f\) is \((C + ε)\)-surjective onto the topological closure of \(K\) for every positive \(ε\).

Let \(f : M_0 → M_1\) and \(g : M_1 → M_2\) be bounded maps between normed groups. Assume there are positive constants \(C\) and \(D\) such that:

\(f\) is \(C\)-surjective onto \(\ker g\).

\(g\) is \(D\)-surjective onto its image.

Then for every positive \(ε\), \(\widehat{f}\) is \((C + ε)\)-surjective onto \(\ker \widehat{g}\).

Let \(M\) be a profinitely filtered pseudo-normed group with action of \(T^{-1}\). For any \(r ∈ (0, 1)\), any \(r\)-normed \(\mathbb Z[T^{\pm 1}]\)-module \(V\), any \(c{\gt}0\) and any \(a\), the map

has norm bounded by \(r^{-1}+1\) and is \(\frac{r}{1-r}(1+\epsilon )\)-surjective.

A *polyhedral lattice* is a finite free abelian group \(\Lambda \) equipped with a norm \(‖\cdot ‖_\Lambda \colon \Lambda \otimes \mathbb R\to \mathbb R\) such that there exists a finite set \(\{ \lambda _1, \dots , \lambda _n\} \subset \Lambda \) that generate the norm: that is to say, for every \(\lambda \in \Lambda \) there exist \(c_1, \dots , c_n \in \mathbb Q\) such that \(\lambda = \sum c_i \lambda _i\) and \(\| \lambda \| = c_i\| \lambda _i\| \).

Equivalently (but not verified in Lean): the norm is given by the supremum of finitely many linear functions on \(\Lambda \); or once more, equivalently, the “unit ball” \(\{ \lambda \in \Lambda \otimes \mathbb R\mid ‖\lambda ‖_\Lambda \leq 1\} \) is a polyhedron.

Let \(M\) be a profinitely filtered pseudo-normed group with \(T^{-1}\)-action that is \(N\)-splittable with error term \(d \ge 0\). Let \(k \ge 1\) be a real number, and let \(c_0 {\gt} 0\) satisfy \(d \le \frac{(k - 1) c_0}{N}\). For every \(c\), consider the Cech nerve of the summation map \(M^N_{c/N} \to M_c\). By applying the functor \(\hat V(\_ )\) and taking the alternating face map complex, we obtain a system of complexes

This system of complexes is weakly \(\le k\)-exact in degrees \(\le m\) and for \(c \ge c_0\) with bound \(1\).

Let \(N\) and \(n\) be natural numbers. Then \(\pi ^N_n\) denotes the universal map from exponent \(N \cdot n\) to \(n\) given by \(\sum _{i {\lt} N} [\pi '_{N,i} \otimes n]\).

(On \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) this map is the formal sum of the maps \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) induced by the projection maps \(A^{N \cdot n} = (A^n)^N \to A^n\).)

A *pseudo-normed group* is an abelian group \((M,+)\), together with an increasing filtration \(M_c \subseteq M\) of subsets \(M_c\) indexed by \(\mathbb R_{\ge 0}\), such that each \(M_c\) contains \(0\), is closed under negation, and \(M_{c_1} + M_{c_2} \subseteq M_{c_1 + c_2}\). An example would be \(M=\mathbb {R}\) or \(M=\mathbb {Q}_p\) with \(M_c :=\{ x\, :\, |x|\leq c\} \).

A pseudo-normed group \(M\) is *exhaustive* if \(\varinjlim _c M_c = M\).

Let \(\Lambda \) be a polyhedral lattice, and let \(N {\gt} 0\) be a natural number. (We think of \(N\) as being fixed once and for all, and thus it does not show up in the notation below.)

By \(\Lambda '\) we denote \(\Lambda ^N\) endowed with the norm

This is a polyhedral lattice.

There is a canonical isomorphism between the first row of the double complex

and

which identifies the map induced by the diagonal embedding \(\Lambda \to \Lambda ' = \Lambda ^{(1)}\) with the map induced by \(\sigma ^N \colon N \otimes \mathsf{BD}\to \mathsf{BD}\).

Fix an integer \(m\geq 0\) and constants \(k\), \(K\). Then there exists an \(\epsilon {\gt}0\) and constants \(k_0\), \(K_0\), depending (only) on \(k\), \(K\) and \(m\), with the following property.

Let \(M^{p,q}_c\) be a system of double complexes as above, and assume that it is admissible. Assume further that there is some \(k'\geq k_0\) and some \(H{\gt}0\), such that

for \(i=0,\ldots ,m+1\), the rows \(M^{i,q}_c\) are weakly \(\leq k\)-exact in degrees \(\leq m-1\) for \(c\geq c_0\) with bound \(K\);

for \(j=0,\ldots ,m\), the columns \(M^{p,j}_c\) are weakly \(\leq k\)-exact in degrees \(\leq m\) for \(c\geq c_0\) with bound \(K\);

it satisfies the normed spectral homotopy condition for \(m\), \(H\), \(c_0\), and \(\epsilon \).

Then the first row is weakly \(\leq k'^2\) exact in degrees \(\leq m\) for \(c\geq c_0\) with bound \(2K_0H\).

We say that the system of double complexes \(M^{p,q}_c\) satisfies the *normed spectral homotopy condition* for \(m \in \mathbb {N}\) and \(H, c_0, \epsilon \in \mathbb {R}_{\ge 0}\) if the following condition is satisfied:

For \(q=0,\ldots ,m\) and \(c\geq c_0\), there is a map \(h^q_{k'c} \colon M^{0,q+1}_{k'c}\to M^{1,q}_c\) with

for all \(x\in M^{0,q+1}_{k'c}\), and such that for all \(c\geq c_0\) and \(q=0,\ldots ,m\) the “homotopic” map

factors as a composite of the restriction \(\mathrm{res}_{k'^2c,c}^{0,q}\) and a map

that is a map of complexes (in degrees \(\leq m\)), and satisfies the estimate

for all \(x\in M^{0,q}_c\).

Let \(N\) and \(n\) be natural numbers. Then \(\sigma ^N_n\) denotes the universal map from exponent \(N \cdot n\) to \(n\) given by \([\sum _{i {\lt} N} \pi '_{N,i} \otimes n]\).

(On \(\mathbb {Z}[A^{N \cdot n}] \to \mathbb {Z}[A^n]\) this map is induced by the summation map \(A^{N \cdot n} = (A^n)^N \to A^n\).)

A *system of complexes* of normed abelian groups is for each \(c \in \mathbb R_{\ge 0}\) a complex

of normed abelian groups together with maps of complexes \(\mathrm{res}_{c',c}: C_{c'}^\bullet \to C_c^\bullet \), for \(c' ≥ c\), satisfying \(\mathrm{res}_{c,c}=\mathrm{id}\) and the obvious associativity condition. In other words, a functor from \((\mathbb R_{\ge 0})^{\mathrm{op}}\) to cochain complexes of semi-normed groups.

Let \(f\) be a universal map from exponent \(m\) to \(n\). Let \(c_1, c_2 \in \mathbb R_{\ge 0}\). We say that \((c_1, c_2)\) is *\(f\)-suitable*, if for all basic universal maps \(g\) that occur in the formal sum \(f\), the pair of nonnegative reals \((c_1, c_2)\) is \(g\)-suitable.

Let \(f\) be a universal map and let \(r, r', c_1, c_2 \in \mathbb R_{\ge 0}\). We say that \((c_1, c_2)\) is *very suitable* for \((f, r, r')\) if there exist \(N, b \in \mathbb {N}\) and \(c' \in \mathbb {R}_{\ge 0}\) such that:

\(f\) is bound by \(N\) (see Definition 1.1.3)

\((c_1, c')\) is \(f\)-suitable

\(r ^b N ≤ 1\)

\(c' ≤ (r') ^b c_2\)

Let \(k \ge 1\), \(c_0 \ge 0\) be real numbers, and \(m \in \mathbb N\). Let \(C_\bullet ^\bullet \) be a system of complexes, and for each \(c \ge 0\) let \(D_c\) be a cochain complex of semi-normed groups. Let \(f_c \colon C^\bullet _{kc} \to D^\bullet _c\) and \(g_c \colon D^\bullet _c \to C^\bullet _c\) be norm-nonincreasing morphisms of cochain complexes of semi-normed groups such that \(g_c \circ f_c\) is the restriction map \(C^\bullet _{kc} \to C^\bullet _c\). Assume that for all \(c \ge c_0\) the cochain complex \(D_c\) is normed exact. Then \(C_\bullet ^\bullet \) is weakly \(\le k\)-exact in degrees \(\le m\) and for \(c \ge c_0\) with bound \(1\).

Let \(M^\bullet _\bullet \) and \(M'^\bullet _\bullet \) be two admissible collections of complexes of complete normed abelian groups. For each \(c\geq c_0\) let \(f^\bullet _c: M^\bullet _c\to M'^\bullet _c\) be a collection of maps between these collections of complexes which all commute with all restriction maps. Assume moreover that we are given two constants \(r_1, r_2 \geq 0\) such that:

for all \(i, c\geq c_0\) and all \(x\in M^i_c\)

\[ ‖f(x)‖ ≤ r_1‖x‖; \]for all \(i ≤ m+1, c \geq c_0\) and all \(y\in M'^i_c\), there exists \(x\in M^i_c\) such that

\[ f(x) = y \mbox{ and } ‖x‖ ≤ r_2‖y‖. \]

Let \(N^\bullet _c\) be the collection of kernel complexes, with the induced norm; this is again an admissible collection of complexes.

Assume that \(M^\bullet _c\) (resp. \(M'^\bullet _c\)) is weakly \(\leq k\)-exact (resp. \(≤ k'\)-exact) in degrees \(\leq m\) for \(c\geq c_0\) with bound \(K\) (resp. \(K'\)). Then \(N^\bullet _c\) is weakly \(\leq kk'\)-exact in degrees \(\leq m-1\) for \(c\geq c_0\) with bound \(K + r_1r_2KK'\).

Let \(M^\bullet _\bullet \) be an admissible collection of complexes of complete normed abelian groups.

Assume that \(M^\bullet _c\) is weakly \(\leq k\)-exact in degrees \(\leq m\) for \(c\geq c_0\) with bound \(K\). Then \(M^\bullet _c\) is \(\leq k^2\)-exact in degrees \(\leq m\) for \(c\geq c_0\) with bound \(K+δ\), for every \(δ {\gt} 0\).

Let \(M^\bullet _\bullet \) and \(M'^\bullet _\bullet \) be two admissible collections of complexes of complete normed abelian groups. For each \(c\geq c_0\) let \(f^\bullet _c: M^\bullet _c\to M'^\bullet _c\) be a collection of maps between these collections of complexes that are norm-nonincreasing and which all commute with all restriction maps, and assume that there exists these maps satisfy

for all \(i ≤ m+1\) and all \(x\in M^i_{k''c}\). Let \(N^\bullet _c=M'^\bullet _c/M^\bullet _c\) be the collection of quotient complexes, with the quotient norm; this is again an admissible collection of complexes.

Assume that \(M^\bullet _c\) (resp. \(M'^\bullet _c\)) is weakly \(\leq k\)-exact (resp. \(≤ k'\)-exact) in degrees \(\leq m\) for \(c\geq c_0\) with bound \(K\) (resp. \(K'\)). Then \(N^\bullet _c\) is weakly \(\leq kk'k''\)-exact in degrees \(\leq m-1\) for \(c\geq c_0\) with bound \(K'(KK'' + 1)\).