Proof. For \(x=(x_0,\ldots,x_j)\) put \(s_a=\sum_{b<a}x_b\), so that \(s_0=0\) and \(s_{j+1}=1\). For \(1\leq r\leq\ell\) define \[ \rho_r(x)_a =\ell\max\left\{0, \min\left(s_{a+1},\frac r\ell\right) -\max\left(s_a,\frac{r-1}\ell\right)\right\}, \qquad 0\leq a\leq j. \tag{1}\] This is \(\ell\) times the length of the intersection of \([s_a,s_{a+1}]\) with the \(r\)th interval of length \(1/\ell\). The coordinates are continuous, nonnegative, and sum to one.

Every nonempty fiber of a monotone map \(\alpha:[j]\to[k]\) is a consecutive block. Passing from \(x\) to \(\alpha_*x\) merges the adjacent mass intervals in that block. The intersection length of the merged interval with a fixed slice equals the sum of its constituent intersection lengths. Empty fibers contribute zero. It follows that \(\alpha_*\rho_r(x)=\rho_r(\alpha_*x)\) for every \(r\). This verifies naturality for all maps of \(\Delta\), including its surjections.

A nonvertex value \(\rho_r(x)\) requires at least one of the internal breaks \(s_1,\ldots,s_j\) to lie strictly between \((r-1)/\ell\) and \(r/\ell\). Distinct such slices have disjoint interiors and therefore require distinct breaks. There are at most \(j\) of them. Breaks lying at slice endpoints do not create nonvertex values. ◻

Proof of 4. Work in the stable functor category \(\mathcal D=\operatorname{Fun}(\Delta,\mathrm{Sp})\). Its equivalences, cofibers, and smash products are computed objectwise. Consider the cosimplicial commutative ring spectrum \[B^j=R^{\wedge(j+1)}.\] For a monotone map \(\alpha:[j]\to[k]\), its structure map multiplies the factors in each fiber of \(\alpha\) and inserts a unit for each empty fiber. This describes its cofaces and codegeneracies, together with their coherences. In particular, the maps \((B^j)^{\wedge\ell}\to B^j\) given by multiplication are natural in \([j]\).

The nullhomotopy on the vertices. The functor \(V\) is the covariant representable \(\operatorname{Hom}_\Delta([0],-)\). Thus \(\Sigma^\infty(G\wedge V^\bullet_+)\) is the left Kan extension of \(\Sigma^\infty G\) from the object \([0]\). The resulting adjunction is an equivalence of mapping spaces \[ \operatorname{Map}_{\mathcal D} \bigl(\Sigma^\infty(G\wedge V^\bullet_+),\Sigma^dB^\bullet\bigr) \simeq \operatorname{Map}_{\mathrm{Sp}}(\Sigma^\infty G,\Sigma^dR). \tag{2}\] Indeed, the left Kan extension in degree \(j\) is the wedge of copies of \(\Sigma^\infty G\) indexed by the \(j+1\) maps \([0]\to[j]\); the adjunction says that a map from this diagram is determined coherently by its value on the copy in degree zero.

There is a natural map \[a:\Sigma^\infty(E\wedge\Delta^\bullet_+) \longrightarrow\Sigma^dB^\bullet\] which forgets the simplex coordinate and applies \(v\) followed by the unit. Its restriction to \(G\wedge V^\bullet_+\) corresponds under [eq:vertices-yoneda] to the assumed null map into \(\Sigma^dR\). Choose a nullhomotopy there. The mapping-space adjunction supplies a single natural nullhomotopy on the vertex diagram. The cofiber universal property therefore gives a map \[\psi:Q^\bullet\longrightarrow\Sigma^dB^\bullet, \qquad Q^j=\Sigma^\infty\left( \frac{E\wedge\Delta^j_+}{G\wedge V^j_+}\right),\] and a homotopy \(\psi q\simeq a\) in \(\mathcal D\), where \(q\) is the quotient map. The displayed ordinary quotients compute the derived cofibers: the inclusion of the indicated pointed CW subspace is a cofibration in each degree. This also permits us to use the actual collapse maps in the next step.

Collapse on the whole union. Put \(U=U_{\ell,n}\). The maps of 5 give a natural map of pointed spaces \[U\wedge\Delta^j_+\longrightarrow(E\wedge\Delta^j_+)^{\wedge\ell}, \quad (x_1\wedge\cdots\wedge x_\ell,x) \longmapsto \bigwedge_{r=1}^\ell(x_r,\rho_r(x)).\] Follow it by the smash of the quotient maps to obtain, after suspension, a map into \((Q^j)^{\wedge\ell}\). If \(j<n\), at least \(n\) of the factors \(x_r\) belong to \(G\), while at most \(j\) of the simplex parameters \(\rho_r(x)\) are nonvertices. Some factor is consequently in \(G\wedge V^j_+\) and is collapsed. The resulting map \[ \Sigma^\infty(U\wedge\Delta^j_+) \longrightarrow(Q^j)^{\wedge\ell} \tag{3}\] is the zero map for every \(j<n\). It is the zero natural transformation on these degrees, since it is already a literal collapse on the diagrams of spaces.

Compose [eq:sliced-union-collapse] with \(\psi^{\wedge\ell}\) and multiplication to obtain a natural map into \(\Sigma^{d\ell}B^j\). Smashing the chosen factorization homotopy \(\psi q\simeq a\) and then multiplying identifies this map with the map which forgets the simplex coordinate and applies \(v^{\wedge\ell}|_U\) followed by the unit. This identification is natural: each map \(a\) forgets its own simplex parameter, so it is independent of the particular values of \(\rho_r(x)\).

The natural projection \[\Sigma^\infty(U\wedge\Delta^\bullet_+) \longrightarrow\operatorname{const}(\Sigma^\infty U)\] is an objectwise equivalence, since every \(\Delta^j\) is contractible. Invert this equivalence in the functor category and restrict to the full subcategory \(\Delta_{\leq n-1}\). We have proved that the cone \[\operatorname{const}(\Sigma^\infty U) \longrightarrow\operatorname{const}(\Sigma^{d\ell}\mathbb S) \longrightarrow\Sigma^{d\ell}B^\bullet\] is null on that subcategory. Inverting an objectwise equivalence here does not require a natural choice of a simplex vertex or a natural contraction of the simplices.

Restriction to the unit cube. The rule \[S\longmapsto[|S|-1]\] defines a functor from the punctured cube \(\mathcal P_n^\times\) to \(\Delta_{\leq n-1}\): an inclusion of subsets induces the ordered injection on their enumerations. Restricting \(B^\bullet\) gives the diagram \(S\mapsto R^{\wedge|S|}\), whose maps insert units. The null cone just constructed therefore yields a null composite \[ \Sigma^\infty U\xrightarrow{v^{\wedge\ell}} \Sigma^{d\ell}\mathbb S \longrightarrow \Sigma^{d\ell}\operatorname*{holim}_{S\in\mathcal P_n^\times} R^{\wedge|S|}. \tag{4}\] Only restriction of cones is used; no assertion about cofinality or an identification of partial totalizations is needed.

Adjoin the value \(\mathbb S\) at the empty subset. The resulting \(n\)-cube is the external smash product of \(n\) copies of \(\mathbb S\to R\). Its total fiber is \(I_R^{\wedge n}\). To verify this, compute fibers successively in its \(n\) directions. Taking the fiber in the last direction replaces every map \(C\wedge\mathbb S\to C\wedge R\) by \(C\wedge I_R\), because smash with \(C\) preserves finite fiber sequences in spectra. Induction on the remaining directions gives \(I_R^{\wedge n}\). Successive fibers compute the fiber of the initial vertex over the limit of the punctured cube. For two directions the relevant identity is \[\operatorname{fib}(A\longrightarrow B\times_D C) \simeq \operatorname{fib}\bigl( \operatorname{fib}(A\to B)\longrightarrow \operatorname{fib}(C\to D)\bigr),\] obtained by expressing each fiber as a pullback over zero. Repeated application gives the assertion for any number of directions, because finite limits commute. Thus there is a fiber sequence \[I_R^{\wedge n}\xrightarrow{\varphi_{R,n}}\mathbb S \longrightarrow \operatorname*{holim}_{S\in\mathcal P_n^\times}R^{\wedge|S|}.\] The chosen nullhomotopy in [eq:unit-cube-null-cone] gives the required lift through its suspended fiber. ◻

Proof. For a single cell in degree \(s\leq D\), the mapping group is \([S^s,Y]=\pi_sY=0\). Attaching finitely many such cells proves the first assertion by induction: the cofiber of an attaching step is a finite wedge of these spheres, and the exact sequence obtained by mapping into \(Y\) makes the mapping group of the enlarged complex zero when those of the preceding complex and the cofiber are zero. For the second assertion, apply the first to \(Y=\tau_{\geq k}\mathbb S_{(p)}\) and use the exact sequence obtained from the fiber sequence \[\tau_{\geq k}\mathbb S_{(p)}\longrightarrow\mathbb S_{(p)} \longrightarrow\tau_{<k}\mathbb S_{(p)}.\] ◻

Proof. The unit-preserving retract follows, for example, from the \(\mathbb E_4\) splitting of \(MU_{(p)}\) in (Basterra and Mandell 2013, Theorem 1.1, the following paragraph, and Section 5). We use the underlying retract of spectra for ordinary homology and the forward map under \(\mathbb S_{(p)}\) to transfer unit fibers.

Let \(C\) be the cofiber of \(\mathbb S_{(p)}\to BP\). The mod-\(p\) homology calculation in (Basterra and Mandell 2013, sec. 5) gives \[H_0(BP;\mathbb F_p)=\mathbb F_p,\qquad H_i(BP;\mathbb F_p)=0\quad(0<i<2p-2).\] The unit induces the degree-zero isomorphism, and \(BP\) is connective. Its unit cofiber therefore has zero mod-\(p\) homology in every degree below \(2p-2\).

The splitting above exhibits \(BP\) as a retract of \(MU_{(p)}\). The ordinary integral homology of \(MU\) is \(\mathbb Z[b_1,b_2,\ldots]\), with \(|b_i|=2i\) (Lurie 2010, Lecture 7, Proposition 2 and Corollary 3); in particular, it is finitely generated in every degree. Taking the retract and then the homology sequence of the unit cofiber shows that \(H_i(C;\mathbb Z_{(p)})\) is finitely generated in every degree. For each \(i<2p-2\), the universal coefficient sequence contains an injection \[H_i(C;\mathbb Z_{(p)})/pH_i(C;\mathbb Z_{(p)}) \lhook\joinrel\longrightarrow H_i(C;\mathbb F_p)=0.\] Nakayama’s lemma therefore gives \(H_i(C;\mathbb Z_{(p)})=0\) for every \(i<2p-2\). The spectrum \(C\) is connective. If it had a nonzero homotopy group below degree \(2p-2\), let \(b\geq0\) be the least such degree. The stable Hurewicz isomorphism in the first nonzero degree would give \(\pi_b(C)\cong H_b(C;\mathbb Z_{(p)})=0\), a contradiction. Thus \(C\) is \((2p-2)\)-connective, and \(I_{BP}\simeq\Sigma^{-1}C\) is \((2p-3)\)-connective.

The map under the sphere induces a map \((I_{MU})_{(p)}\to I_{BP}\) commuting with the fiber inclusions. Smashing \(n\) times proves the last assertion, since smash products add connectivity bounds. ◻

Proof of 1. Put \(e=\dim E\), \(n_\ell=\lceil\ell/2\rceil\), and \[K_\ell=\Sigma^{-d\ell}\Sigma^\infty U_\ell.\] The finite subcomplex \(U_\ell\subset E^{\wedge\ell}\) has cell dimension at most \(e\ell\), so \(K_\ell\) has upper cell degree \[ D_\ell=(e-d)\ell. \tag{7}\] The class under consideration is now a map \(\beta_\ell:K_\ell\to\mathbb S\). By 4, it lifts through \(I_{MU}^{\wedge n_\ell}\).

If \(e<d\), 7 applied directly to the connective sphere shows that \(\beta_\ell=0\) for every \(\ell\). Assume henceforth that \(e\geq d\) and put \[k_\ell=(e-d)\ell+1.\] For a fixed prime \(p\), localization of the lift gives a lift through \(I_{MU_{(p)}}^{\wedge n_\ell}\). If \(e>d\), 6 gives \[f_{MU_{(p)}}(k_\ell)=o(\ell),\] so that \(n_\ell\geq f_{MU_{(p)}}(k_\ell)\) for all sufficiently large \(\ell\). If \(e=d\), the same inequality eventually holds because \(k_\ell=1\) is fixed and \(n_\ell\to\infty\). By [eq:nilpotence-truncated-composite], the composite of \((\beta_\ell)_{(p)}\) with \(\tau_{<k_\ell}\mathbb S_{(p)}\) is zero. By 7 and [eq:filtered-product-cell-bound], \((\beta_\ell)_{(p)}=0\). This proves eventual vanishing for each fixed prime.

To obtain a single threshold, choose an integer \(P\geq2\) such that \[ \frac{2p-3}{2}>\max(e-d,0) \qquad\text{for every prime }p>P. \tag{8}\] For such a prime, 8 turns the lift of \((\beta_\ell)_{(p)}\) into a lift through \(I_{BP}^{\wedge n_\ell}\). Its target has connectivity at least \[n_\ell(2p-3)\geq\frac\ell2(2p-3)>(e-d)\ell=D_\ell.\] The first assertion of 7 makes this lift zero, for every \(\ell\). Only the finitely many primes at most \(P\) remain. Take the maximum of their eventual thresholds from the preceding paragraph. Above that maximum, \(\beta_\ell\) vanishes after localization at every prime.

Because \(K_\ell\) is finite, localization of its stable mapping group agrees with maps from its localization. One way to see this is the duality identity \[F(K_\ell,\mathbb S)\simeq D K_\ell, \qquad (D K_\ell)_{(p)}\simeq F((K_\ell)_{(p)},\mathbb S_{(p)}),\] where \(D K_\ell\) denotes the finite Spanier–Whitehead dual. An element of an abelian group which vanishes in every localization at a prime is zero. Indeed, an element of infinite order remains nonzero in every such localization, and a nonzero element of finite order remains nonzero at any prime dividing its order. Applying this to \([K_\ell,\mathbb S]\) gives \(\beta_\ell=0\) integrally, as required. ◻

Proof. All restriction maps are induced by maps of the defining pairs. For the subpolyhedron assertion, triangulate \(P\times Z'\) so that \(P\times Z\) and \(A_c(f,Z')\) are subpolyhedra. Collapsing the latter identifies no two points of the former except those already collapsed in \(Q_c(f,Z)\). Quotients of this finite polyhedral pair give the assertion, including the stated basepoint convention.

For products the relative subset is exactly the union of the subsets on which at least one factor belongs to \(A_c(f_i,Z_i)\). Its quotient is the smash product of the individual quotients. Choosing all collapses as products of the same one-dimensional collapse makes the identity of maps literal. Composing with units and multiplying in \(MU\) gives the last assertion. ◻

Proof. The empty and full arrangements are immediate. In the remaining case, for \(y\notin Z\) set \[\lambda(y)=\max\{1,\epsilon/d_Z(y)\}, \qquad \lambda_u(y)=1+u(\lambda(y)-1),\quad 0\le u\le1.\] Define a coordinatewise homotopy by \[ H_u(y)_i=\operatorname{sgn}(y_i) \min\{\lambda_u(y)|y_i|,\max\{|y_i|,\epsilon\}\}, \tag{12}\] with value zero when \(y_i=0\). It is continuous on \(T\setminus Z\): \(\lambda_u\) is locally bounded there, which also gives continuity at zero coordinates. Coordinates of magnitude at least \(\epsilon\) are fixed. Each smaller coordinate is expanded, without changing sign, by at most enough to reach magnitude \(\epsilon\).

Every coordinate interval of a box either contains \([-\epsilon,\epsilon]\) or lies outside that interval. Thus [eq:coordinate-expansion] preserves even asymmetric boxes. Coordinate magnitudes never decrease, and zero coordinates remain zero, so the homotopy stays in the complement and preserves \(V_\alpha(Z)\) at every time. In fact it fixes that subspace pointwise, since \(d_Z(y)\ge\epsilon\) implies \(\lambda(y)=1\). If \(d_Z(y)<\epsilon\), each set \(J\) contains an index with \(|y_i|\ge d_Z(y)\), and that coordinate has magnitude at least \(\epsilon\) at time one. Hence \(d_Z(H_1(y))\ge\epsilon\). This proves the retraction. The formula is the same for every box. In particular it applies to any compact family in the unthickened complement, with no need to move arbitrarily small nonzero coordinates discontinuously to a fixed radius.

For connectivity, first move a compact family from the closed box into its interior without changing its zero coordinates: shrink intervals containing zero by a fixed factor in \((0,1)\), and move the other intervals toward an interior point of the same sign. A map from \(S^i\) into the complement, \(i\le q-2\), can then be approximated by a PL map in the interior and extended over a triangulated \(D^{i+1}\) into the convex box. A sufficiently small generic perturbation of its vertices misses all the excluded subspaces, since \[(i+1)+(M-q)<M.\] The perturbed boundary map is homotopic to the original within the complement. This proves the claimed connectivity, including path connectedness in the applicable range. Existence of a point in the complement when \(q\ge1\) follows from the same dimension observation. The retraction transfers connectivity to \(V_\alpha(Z)\). ◻

Proof. Choose one \(\rho>0\) with \[ \rho<\epsilon,\qquad \rho<\min_\alpha \operatorname{dist}_\infty \bigl(B_\alpha,T\setminus\operatorname{int}B_\alpha^+\bigr). \tag{16}\] All minima are positive, by finiteness and strict containment. The formula \((y,z)\mapsto\kappa_\rho(y-z)\) descends to a pairing \[ \mathcal C_\alpha(Z)\wedge\Sigma^\infty T_\alpha(Z) \longrightarrow\Sigma^M\mathbb S. \tag{17}\] Indeed, if \(y\in V_\alpha(Z)\) and \(z\in Z\), their distance is at least \(\epsilon\). If \(z\notin\operatorname{int}B_\alpha^+\) and \(y\in B_\alpha\), the second inequality in [eq:common-collapse-radius] applies. These are exactly the two collapsed subspaces. Since \(V_\alpha(Z)\) is a subpolyhedron, its ordinary quotient by inclusion in \(B_\alpha\) represents the cofiber in [eq:local-cofiber].

First take \(Z=L_J\), with \(r=|J|\) and \(b=M-r\). If it misses \(B_\alpha\), some required coordinate is bounded away from zero by more than \(\epsilon\), so \(V_\alpha(Z)=B_\alpha\). It also misses \(B_\alpha^+\), and both sides of [eq:box-duality] are zero. Otherwise split the box into its normal and tangential factors, \(B_\alpha=B_N\times B_T\). Then \[B_\alpha/V_\alpha(L_J) \cong \left(B_N/(B_N\setminus(-\epsilon,\epsilon)^r)\right) \wedge(B_T)_+\simeq S^r, \qquad T_\alpha(L_J)\cong S^b.\] The conventions include \(r=0\) and \(b=0\). Fixing an interior point \(w\in B_T\) identifies the pairing with the collapse of \((u,z_T)\mapsto(u,w-z_T)\), up to a coordinate permutation. It has degree \(\pm1\): the origin has one preimage, and its local linear map is \(\operatorname{diag}(I_r,-I_b)\). Its adjoint is therefore an equivalence \(S^r\simeq F(S^b,\Sigma^M\mathbb S)\).

Here is the excision argument for unions. For coordinate planes, \[U_\epsilon(L_J)\cap U_\epsilon(L_K) =U_\epsilon(L_{J\cup K}).\] Distributing finite unions gives, for arrangements \(Z,Z'\), the identities \[\begin{align*} V_\alpha(Z\cup Z')&=V_\alpha(Z)\cap V_\alpha(Z'),\\ V_\alpha(Z\cap Z')&=V_\alpha(Z)\cup V_\alpha(Z'). \end{align*}\] A rectangular subdivision followed by triangulation makes all these closed complements subpolyhedra. Their union square is a homotopy pushout. Applying suspension spectra and cofibers into the constant spectrum \(\Sigma^\infty(B_\alpha)_+\) gives the homotopy pullback description \[ \mathcal C_\alpha(Z\cup Z')\simeq \mathcal C_\alpha(Z) \mathop{\times}\limits^{h}_{\mathcal C_\alpha(Z\cap Z')} \mathcal C_\alpha(Z'). \tag{18}\] We used here that a pushout of spectra is also a pullback. On the test-space side there is a pushout of pointed polyhedra along cofibrations \[T_\alpha(Z\cup Z')\cong T_\alpha(Z)\cup_{T_\alpha(Z\cap Z')}T_\alpha(Z').\] Its function spectra give the same pullback as in [eq:cofiber-arrangement-excision], and the pairings commute with these maps. Induction on the number of planes proves [eq:box-duality]: the intersection of one plane with a union of fewer planes is again a union of no more than that many coordinate planes. The empty arrangement contributes zero throughout.

Finally, for an arrangement inclusion \(Z\subseteq Z'\), the maps are \[\mathcal C_\alpha(Z')\longrightarrow\mathcal C_\alpha(Z), \qquad T_\alpha(Z)\longrightarrow T_\alpha(Z').\] For a box inclusion indexed by \(\alpha\le\beta\), the maps are \[\mathcal C_\alpha(Z)\longrightarrow\mathcal C_\beta(Z), \qquad T_\beta(Z)\longrightarrow T_\alpha(Z),\] the second being quotient collapse. On representatives, each required compatibility says that the pairing is \(\kappa_\rho(y-z)\). If the smaller test quotient additionally collapses \(z\), the pairing with \(y\in B_\alpha\) was already the basepoint by [eq:common-collapse-radius]. Thus the adjoints are natural equivalences of diagrams, not just objectwise identifications. ◻

Proof. The empty domain or empty arrangement needs no argument. Otherwise the dimension hypotheses imply \(q\ge3\).

Choice of boxes. Subdivide \(P\) so finely that \(f(\sigma)\) has maximum-norm diameter less than \(c\) for each simplex. Thicken its coordinate bounding box by a radius between \(2c\) and \(3c\), increasing strictly with the dimension of \(\sigma\). A face box is then contained in the interior of its coface box. Choose the finitely many radii generically so that no endpoint is zero. The resulting boxes \(B_\sigma\) have the properties \[ \operatorname{dist}_\infty \bigl(f(\sigma),T\setminus\operatorname{int}B_\sigma\bigr)>c, \qquad \sup_{x\in\sigma,\,y\in B_\sigma}\|y-f(x)\|_\infty<\tfrac12. \tag{19}\] They all lie in \((-2,2)^M\); indeed the latter supremum is less than \(4c\). Enlarge them by one sufficiently small uniform amount to obtain \(B_\sigma^+\), preserving the coordinate incidences and all inclusions. Choose a common \(\epsilon\) and then a common collapse radius \(\rho<c\) as in [lem:coordinate-thickening,lem:box-duality]. Write \(V_\sigma=V_\sigma(Z)\) and \(T_\sigma=T_\sigma(Z)\).

The stable lift as a map of diagrams. Index covariant diagrams by the nonempty simplices of \(P\), with maps from faces to cofaces. Set \[\mathcal D_\sigma=\Sigma^\infty\sigma_+, \qquad \mathcal B_\sigma=\Sigma^\infty(B_\sigma)_+, \qquad \mathcal V_\sigma=\Sigma^\infty(V_\sigma)_+.\] The maps \(f|_\sigma\) define a natural map \(\mathcal D\to\mathcal B\). Its obstruction to lifting through \(\mathcal V\to\mathcal B\) is the composite to the cofiber diagram \(\mathcal C\).

For clarity, the source cell diagram has latching maps \[ \underset{\tau\subsetneq\sigma}{\operatorname{colim}}\tau_+ =\partial\sigma_+\longrightarrow\sigma_+. \tag{20}\] They are cofibrations; at a vertex the latching object is the basepoint. Consequently this is a cofibrant diagram, and compatible mapping out of its suspension spectrum computes the derived mapping spectrum. This fact can also be seen directly: add the simplices in increasing dimension. At each step a compatible map is an extension over \(\sigma\) of the already specified map on \(\partial\sigma\). Restriction along [eq:cell-latching] is a mapping-space fibration after fibrant replacement. Induction therefore makes compatible mapping invariant under objectwise equivalences and preserves the fiber sequences of mapping spectra used here.

The duality in 13 identifies the obstruction with the adjoint of the equality test on \(\sigma_+\wedge T_\sigma\). There is a well-defined based map \[ \sigma_+\wedge T_\sigma\longrightarrow Q_c(f,Z), \qquad [x,z]\longmapsto[x,z]. \tag{21}\] Indeed, if \(z\) lies outside \(\operatorname{int}B_\sigma^+\), then [eq:avoidance-box-control] puts \((x,z)\) in the relative set of the global test. For a face \(\sigma\subset\tau\), compare the two maps on \(\sigma_+\wedge T_\tau\): including \(\sigma\) in \(\tau\) or collapsing \(T_\tau\) to \(T_\sigma\) gives the same based map to \(Q_c(f,Z)\). Every additional collapse already lands in the global relative set by [eq:avoidance-box-control]. After adjunction these equalities give the required natural transformation. Pulling back one stable nullhomotopy of \(\varphi_\rho(f,Z)\) thus gives a compatible nullhomotopy of the obstruction diagram. This uses the assumed relative point-test vanishing, rather than separately chosen local nullhomotopies.

The category of spectrum diagrams is stable. Mapping out of \(\mathcal D\) into the objectwise cofiber sequence \(\mathcal V\to\mathcal B\to\mathcal C\) now gives a stable lift \(\mathcal D\to\mathcal V\), together with a compatible homotopy of its composite to the map given by \(f\).

Augmentation and desuspension. For an unbased space \(V\), the augmentation \(\Sigma^\infty V_+\to\mathbb S\) sends every point to the nonbasepoint of \(S^0\). A map of spaces into \(V\) therefore has augmentation equal to the unit, whereas an arbitrary stable map need not. Each \(B_\sigma\) is contractible, so its augmentation is an equivalence. After adjunction, augmenting the compatible homotopy in the box diagram gives a homotopy from the stable lift’s augmentation to the constant unit map. We retain the lift together with this homotopy by setting \[ E_\sigma=\operatorname{hofib}_{1}\left( \Omega^\infty\Sigma^\infty(V_\sigma)_+ \longrightarrow\Omega^\infty\mathbb S\right), \tag{22}\] where \(1\) is the unit point. Naturality of the augmentations makes these data a single derived map from the cell diagram \(\sigma\) into the homotopy-fiber diagram \(E\). We keep the lift and its augmentation homotopy together when choosing functorial models.

There is a natural comparison \(j_\sigma:V_\sigma\to E_\sigma\). Its augmentation is literally the unit, so the path in the homotopy fiber can be taken constant. To compute its connectivity, choose a point \(v_\sigma\in V_\sigma\). The resulting splitting of the augmentation identifies \(E_\sigma\) with \(\Omega^\infty\Sigma^\infty(V_\sigma,v_\sigma)\) and identifies \(j_\sigma\) with stabilization. The choices are used only for this objectwise connectivity calculation; the comparison itself was defined naturally. By 12, \(V_\sigma\) is \((q-2)\)-connected. Freudenthal therefore makes \(j_\sigma\) at least \((2q-3)\)-connected (Hatcher 2002, Corollary 4.24). In particular, the weaker bound \(2q-4\) is available with room for the stated dimension hypothesis.

Here is an explicit way to realize the lift without making compatible basepoint choices. Replace \(j\) by its natural mapping-path factorization. Objectwise this is \[W_\sigma=\{(v,\gamma):v\in V_\sigma,\ \gamma:[0,1]\to E_\sigma,\ \gamma(0)=j_\sigma(v)\}, \qquad p_\sigma(v,\gamma)=\gamma(1).\] The maps \(p_\sigma\) are fibrations of connectivity at least \(2q-3\). Projection \(W_\sigma\to V_\sigma\) is a natural homotopy equivalence, with section given by constant paths. The cofibrant cell diagram allows the preceding compatible derived map to \(E\) to be represented by an actual map of diagrams, using objectwise fibrant models.

Inductively lift this map to \(W\). At a simplex \(\sigma\), the chosen lifts on proper faces give a map \(\partial\sigma\to W_\sigma\) over the prescribed map \(\sigma\to E_\sigma\). Its extension is possible because the fibers of \(p_\sigma\) are \((2q-4)\)-connected and \(\dim\sigma\le2q-5\). This is the ordinary lifting obstruction for a disk and its boundary. Projection back to \(V\) gives an actual compatible family \(\sigma\to V_\sigma\). No dimension of a separate indexing space is added: the extension domains are exactly the simplices in [eq:cell-latching].

The family glues to a continuous map \(g:P\to(-2,2)^M\setminus Z\). Its image on \(\sigma\) lies in \(B_\sigma\), so [eq:avoidance-box-control] gives the claimed displacement bound. ◻

Proof. If \(K\) is empty the assertion is immediate. Otherwise, choose a continuous \(\chi:[0,1]\to[0,1]\) which is zero on \([0,3/4]\) and equals one at \(1\). Define slotwise \[g'_j(x)=\bigl(1-\chi(\|f_j(x)\|_\infty)\bigr)g_j(x) +\chi(\|f_j(x)\|_\infty)f_j(x).\] Whenever the cutoff is nonzero, one scalar coordinate of \(f_j(x)\) has magnitude greater than \(3/4\). The corresponding coordinate of \(g_j(x)\) has the same sign and magnitude greater than \(1/4\). The convex interpolation consequently cannot turn that slot into zero. All other slots are unchanged. Hence \(g'\) still has at least \(s+1\) nonzero slots everywhere, and it already agrees with \(f\) at each original boundary slot.

The \((s+1)\)-st largest of the finitely many continuous functions \(\|g'_j(x)\|_\infty\) is continuous and positive on \(K\). By compactness choose \(0<\delta<1\) smaller than its minimum. Set \[G_j(x)=\frac{g'_j(x)}{\max\{\delta,\|g'_j(x)\|_\infty\}}.\] This is continuous even at zero, belongs to the unit cube, and sends every slot of radius at least \(\delta\) to the boundary. It fixes every slot already of radius one. Both conclusions follow. ◻

Proof. Take the finite pointed pair \[E=Q_c\bigl(f,T(t,n)\bigr),\qquad G=Q_c\bigl(f,Z_{an}(t,n)\bigr)\subset E.\] A common polyhedral subdivision of the domain, sparse subsets, and relative subsets makes this a pointed CW pair. The origin collapse gives a based map \(v:E\to S^{tn}\) whose restriction to \(G\) is zero after applying the unit into \(MU\). By 1, for all sufficiently large \(\ell\) the map \(v^{\wedge\ell}\) is stably null on the union of smash products having at least \(\lceil\ell/2\rceil\) factors in \(G\).

Split a parameter in \(T(t,b)\) into \(\ell\) consecutive groups of \(n\) slots. Every group outside \(Z_{an}(t,n)\) uses strictly more than \(an\) nonzero slots. Consequently a parameter with at most \(H\le an\ell/4\) nonzero slots has fewer than \(\ell/4\) such groups. In particular, at least \(\lceil\ell/2\rceil\) groups belong to \(Z_{an}(t,n)\). By 11, the quotient for the batch test maps into the preceding union and the batch test is the restriction of \(v^{\wedge\ell}\). Pulling back its single stable nullhomotopy proves the assertion. ◻

Proof. The assertion is immediate for an empty \(P\), so assume \(P\) is nonempty and write \(d_P=\dim P\). Fix \(\ell\) as in 16, large enough that \(ab\ge16\), where \[b=n\ell,\qquad H=\lfloor ab/4\rfloor.\] In particular \(H\ge1\). Fix a positive integer \(B\) of padding batches so large that \[ tb(B+1)\ge \max\{20,2\ell d_P\}. \tag{23}\] The initial \(B\) batches will have output \(f^{\times\ell B}\) and no required boundary count.

For \(N\ge1\) and \(0\le s\le N\), let \[Q_s(t,N)=\{y\in C(t,N): \text{at least $s$ slots of $y$ lie in $\partial[-1,1]^t$}\}.\] This is a finite polyhedron: it is a finite union of products in which specified factors are cube boundaries. Its dimension is at most \(tN\).

Suppose \(j-1\) ordinary batches have already been added. Put \[N_{j-1}=b(B+j-1),\qquad s=(j-1)H.\] The previous output lies in \(Q_s(t,N_{j-1})\). To add the next batch we solve a new problem on the finite polyhedron \[ W_j=Q_s(t,N_{j-1})\times P^\ell, \qquad F_j(y,x)=(y,f^{\times\ell}(x)). \tag{24}\] After a product subdivision \(F_j\) is PL. We first show that its sphere point test vanishes on \[Z_{jH-1}\bigl(t,b(B+j)\bigr).\]

Write the sparse parameter as \((z_{\mathrm o},z_{\mathrm n})\), for the old and new slots. On this parameter set the two closed conditions \[\begin{align*} A_j&:\quad z_{\mathrm o}\in Z_{s-1}(t,N_{j-1}),\\ B_j&:\quad z_{\mathrm n}\in Z_{H-1}(t,b) \end{align*}\] cover the whole set: failure of both would give at least \(s+H=jH\) nonzero slots. If \(s=0\), the first set is empty. On \(A_j\), one of the at least \(s\) boundary slots of \(y\) has zero old parameter, and therefore \[ \|y-z_{\mathrm o}\|_\infty\ge1. \tag{25}\] Thus the old relative quotient is at its basepoint there.

Here is the precise gluing of the relative maps. Set \[E_{\mathrm o}=Q_c\bigl(\mathop{\mathrm{id}}_{Q_s(t,N_{j-1})},T(t,N_{j-1})\bigr), \qquad G_{\mathrm n}=Q_c\bigl(f^{\times\ell},Z_{H-1}(t,b)\bigr).\] On the part satisfying \(B_j\) send \(((y,x),(z_{\mathrm o},z_{\mathrm n}))\) to \([y,z_{\mathrm o}]\wedge[x,z_{\mathrm n}]\) in \(E_{\mathrm o}\wedge G_{\mathrm n}\). On the part satisfying \(A_j\) send it to the basepoint. These maps agree on their intersection by [eq:old-test-basepoint], and the closed pasting lemma gives a continuous map on the full domain. It is constant on the relative subset \(\|F_j-(z_{\mathrm o},z_{\mathrm n})\|_\infty\ge c\): on \(B_j\) at least one smash factor is then the basepoint. It therefore descends to a based map \[ Q_c\bigl(F_j,Z_{jH-1}(t,b(B+j))\bigr) \longrightarrow E_{\mathrm o}\wedge G_{\mathrm n}. \tag{26}\] Composing with the smash of the two point tests gives exactly the point test of \(F_j\), by 11; on \(A_j\) both descriptions are constant by [eq:old-test-basepoint]. The new factor is stably null, since its parameter set is contained in the one in 16. Its smash with the old factor is therefore stably null. This proves vanishing on the entire sparse union by one relative factorization.

We check the avoidance range before carrying out the induction. The current ambient dimension and the codimension of the excluded sparse set are \[ D_j=tb(B+j),\qquad q_j=D_j-t(jH-1). \tag{27}\] Indeed \(0\le jH-1<b(B+j)\), so that sparse set has dimension \(t(jH-1)\). Since \(H\le ab/4\) and \(a<1/16\), \[t(jH-1)<\frac a4D_j<\frac18D_j, \qquad q_j>\frac78D_j.\] On the other hand, [eq:padding-domain,eq:padding-choice] give \[\dim W_j\le tb(B+j-1)+\ell d_P <\frac32D_j.\] For all \(j\ge1\) we have \(D_j\ge20\), and hence \[ 2q_j-5>\frac74D_j-5\ge\frac32D_j>\dim W_j. \tag{28}\] Thus 14 applies to the null point test just constructed. It gives an output avoiding \(Z_{jH-1}\) within distance \(1/2\) of \(F_j\). By 15, this output can be changed to a continuous map \[g_j:W_j\longrightarrow Q_{jH}\bigl(t,b(B+j)\bigr)\] that fixes every boundary slot of \(F_j\).

Starting with \(G_0=f^{\times\ell B}\), define recursively \[G_j(\mathbf x,x)=g_j(G_{j-1}(\mathbf x),x), \qquad \mathbf x\in P^{\ell(B+j-1)},\quad x\in P^\ell.\] This is well-defined because \(G_{j-1}\) takes values in the first factor of \(W_j\). It has at least \(jH\) boundary slots. Boundary slots in the new batch are fixed when that batch enters; boundary slots already present in the old output are fixed at every later step. The padding output initially equals the original input. Induction therefore proves the required agreement with every original boundary slot.

Finally the floor in \(H\) and the padding loss have explicit bounds: \[\frac Hb\ge\frac a4-\frac1b\ge\frac{3a}{16}, \qquad \frac{jH}{b(B+j)} \ge\frac23\frac{3a}{16}=\frac a8 \quad(j\ge2B).\] Taking \(u=\ell(B+j)\) for these arbitrarily large \(j\) proves the theorem. The dimension estimate was applied only to \(W_j\); no bound on the dimension of the full original parameter product is needed. ◻

Proof. Begin with a circle embedded in \(\mathbb R^2\) with its outward normal framing. If \(\mathbb T^k\) has been embedded in \(\mathbb R^{k+1}\) with a framed normal line, its inclusion into \(\mathbb R^{k+2}\) has a framed normal two-plane bundle. The boundary of a sufficiently small tubular disk bundle is consequently a smooth embedded hypersurface \(\mathbb T^k\times S^1\) in \(\mathbb R^{k+2}\). Its outward normal is a framing. Iteration produces \(\mathbb T^{2p}\) in \(\mathbb R^{2p+1}\), and translation and rescaling put it in the specified cube.

Choose a smooth degree-one map \(\mathbb T^2\to S^2=\mathbb{CP}^1\); one may collapse the complement of an oriented disk and then take a smooth approximation. The orthogonal complement of the tautological line over \(\mathbb{CP}^1\) is a line subbundle of \(\mathbb C^2\) with first Chern number one. Pull it back to \(\mathbb T^2\), and then to each indicated factor of \(L\). Taking their direct sum gives the stated embedding. The evaluation follows from the product orientation and the Chern number on each factor. ◻

Proof. Put \(M=(2p+1)n\), \(r=pn\), \(D=P\times Z\), and \(d=d_f\). The rank bound will force the pulled-back top \(MU\) Chern class of \(H_n\) to vanish. We need this vanishing on the preimage of a neighborhood of \(L^n\), where multiplication by the normal Thom class will recover the relative point test.

Fix the normal framing of \(L^n\), of real rank \(n\), obtained by taking products in 18. A tubular neighborhood retraction extends \(H_n\) and its orthogonal projection \(\Pi:L^n\to M_{2r}(\mathbb C)\) to a neighborhood of \(L^n\).

We first extend the embedding to the pullback of a whole tube. On the compact closed subset \(\mathcal T(f,Z)\subset D\), compose the assumed embedding with the source projection \(\Pi\circ d\). This represents it by a continuous \(K\)-by-\(2r\) matrix. Extend the real and imaginary parts of its entries to \(D\) by the Tietze extension theorem (Tao 2010, Theorem 1.10.5). On the original test locus its restriction to the source fibers has a uniform positive lower bound on unit vectors, by compactness. This lower bound persists on the pullback of a sufficiently thin tube: otherwise a sequence of unit source vectors over points tending to the central locus would converge to a vector contradicting injectivity there. If the central locus is empty, compactness instead gives a tube with empty pullback. We may thus choose an open tube \(U\) with compact closure in \((-c,c)^M\) and retraction \(\rho:U\to L^n\) such that on \(d^{-1}(U)\) the pulled-back bundle \(d^*\rho^*H_n\) embeds into \(\mathbb C^K\).

We use the complex orientation of \(MU\) and its Chern and Thom classes, with the direct-sum Whitney formula (Lurie 2010, Lecture 5, Proposition 6 and the following discussion; Lecture 6, Remark 7). List the \(r\) line summands of \(H_n\) and write \[x_i=c_1^{MU}(\lambda_i)\in MU^2(L^n),\qquad 1\le i\le r, \qquad \alpha=\prod_{i=1}^r x_i.\] Here the indices include both the slot and the two-torus factor. Each \(x_i\) is pulled back from a two-dimensional torus. Since \(MU\) is connective, \(MU^4(\mathbb T^2)=0\): filtering that torus by its cells reduces this assertion to \(MU^q(\mathrm{pt})=0\) for \(q>0\). Consequently \(x_i^2=0\).

Pull these classes to \(d^{-1}(U)\). If this space is nonempty, the image of the embedded rank-\(r\) bundle has an orthogonal complement \(F\) in \(d^{-1}(U)\times\mathbb C^K\). Thus \(F\) has rank \(K-r<r\). With a formal variable \(v\) recording Chern index, the Whitney formula and the square-zero identities give \[ c_v^{MU}(F) =\prod_{i=1}^r(1+d^*\rho^*x_i\,v)^{-1} =\prod_{i=1}^r(1-d^*\rho^*x_i\,v). \tag{29}\] The coefficient of \(v^r\) on the left is zero by rank. The coefficient on the right is \((-1)^r d^*\rho^*\alpha\). Hence \[ d^*\rho^*\alpha=0\quad\hbox{in }MU^{2r}(d^{-1}(U)). \tag{30}\] The same assertion is automatic if the tube pullback is empty. This uses only the direct-sum Chern formula; no formula for tensor products of lines is involved.

For the support pairs below, relative \(MU\) means reduced cohomology of homotopy cofibers. The collared ambient tube pairs and the final polyhedral pair also admit the literal quotient descriptions used here. Choose a closed smaller tubular disk bundle \(V\) with \(L^n\subset\operatorname{int}V\) and \(V\subset U\). The normal framing gives a Thom class \[\tau\in MU^n(U,U\setminus\operatorname{int}V).\] Indeed the relative quotient is the Thom space of a trivial real rank-\(n\) bundle, whose Thom class is the \(n\)-fold suspension of the unit. This construction works for odd as well as even \(n\) (Lurie 2010, Lecture 5, Example 1). The relative product \[\eta=(\rho^*\alpha)\tau \in MU^M(U,U\setminus\operatorname{int}V)\] extends by excision to a class supported in \(V\) in \(MU^M(\mathbb R^M,\mathbb R^M\setminus\operatorname{int}V)\). Its image in \[ MU^M\bigl(\mathbb R^M, \mathbb R^M\setminus(-c,c)^M\bigr) \cong\widetilde{MU}^M(S^M) \cong\pi_0MU=\mathbb Z \tag{31}\] is a generator, up to the sign of the normal orientation. To check the integer, apply the ordinary orientation map \(MU\to H\mathbb Z\). It induces an isomorphism on the group in [eq:ambient-point-group]. The ordinary image of \(\eta\) has evaluation \[\left\langle\prod_{i=1}^r c_1(\lambda_i),[L^n]\right\rangle=1\] by 18 and the product orientation. Thus the image of \(\eta\) is exactly the \(MU\) origin class up to sign.

We give the pullback with its relative sets to retain this supported information. On \(d^{-1}(U)\) the pullback of \(\eta\) vanishes, because its absolute factor is zero by [eq:euler-tube-zero]. Excision for the smaller closed tube therefore gives zero in \[MU^M\bigl(D,D\setminus d^{-1}(\operatorname{int}V)\bigr).\] This excision is valid after pullback: the closure of \(D\setminus d^{-1}(U)\) lies in the interior of \(D\setminus d^{-1}(\operatorname{int}V)\), since \(V\subset U\) and \(V\) is compact. In particular the excision isomorphisms commute with \(d\) and with the relative products used above. As \(V\subset(-c,c)^M\), the relative sets satisfy \[A_c(f,Z)\subset D\setminus d^{-1}(\operatorname{int}V).\] The induced map on relative cohomology sends this zero class to \[ MU^M\bigl(D,\{(x,z):\|f(x)-z\|_\infty\ge c\}\bigr). \tag{32}\] The image is, up to sign, the pullback of the origin class in [eq:ambient-point-group], so it is the \(MU\) point test and is zero. Finally the pair in [eq:point-test-relative-mu] is a finite polyhedral pair: \(f\) is PL, \(Z\) is a finite union of coordinate cubes, and the relative condition is a finite union of linear inequalities after subdivision. Its relative cohomology is therefore the reduced cohomology of the exact quotient in 10. This proves the required vanishing. ◻

Proof. If \(Y\) is empty, take \(P\) empty and the unique maps. Otherwise put \(M=(2p+1)n\) and \(r=pn\). Represent the embedding on the compact test locus by a \(K\)-by-\(2r\) matrix \(A\), by first projecting onto the fixed source bundle. Extend its entries to \(Y\times Z\) by the Tietze theorem (Tao 2010, Theorem 1.10.5). Choose \(\delta>0\) such that on the central test locus \[ \|A(y,z)v\|\ge4\delta\|v\| \quad\hbox{for }v\in(H_n)_{f(y)-z}. \tag{33}\] Compactness of its unit sphere bundle supplies this number. If the locus is empty the condition is vacuous and any \(\delta>0\) may be used.

The real algebra on \(Y\times Z\) generated by continuous real functions on \(Y\) and the Euclidean coordinates of \(z\) contains the constants and separates points. By the real Stone–Weierstrass theorem (Tao 2010, Theorem 1.10.18), the real and imaginary entries of \(A\) can be uniformly approximated by elements of this algebra. Only finitely many functions on \(Y\) occur in these approximants. Rescale them into \([-1,1]\) and denote them by \(\varphi_1,\ldots,\varphi_q\). Include every coordinate of \(f\) explicitly, by setting \[j_0(y)=(f(y),\varphi_1(y),\ldots,\varphi_q(y)) \in W=[-1,1]^{M+q}.\] The approximating matrix has the form \(B(j_0(y),z)\), where \(B\) is a matrix of polynomials on \(W\times Z\). Choose the approximation so that its operator-norm error is less than \(\delta\) everywhere on \(Y\times Z\). By [eq:original-injectivity], \[ \|B(j_0(y),z)v\|\ge3\delta\|v\| \quad\hbox{on the original slot bundle}. \tag{34}\]

Let \(C=j_0(Y)\) and let \(f_0:W\to[-1,1]^M\) be projection onto the first \(M\) coordinates. The bundle over a point \((u,z)\) with \(f_0(u)-z\in L^n\) depends only on \(f_0(u)-z\). Consequently [eq:coordinate-injectivity] gives the same bound at every such point of \(C\times Z\), regardless of the choice of a preimage of \(u\) in \(Y\).

For completeness, consider the following compact bad set: \[\begin{split} \mathcal B=\{(u,z,v):\;&u\in W,\ z\in Z,\ \|v\|=1, \ f_0(u)-z\in L^n,\\ &\Pi(f_0(u)-z)v=v, \ \|B(u,z)v\|\le2\delta\}. \end{split}\] Here \(\Pi\) is the continuous orthogonal projection onto \(H_n\). The projection of \(\mathcal B\) onto \(W\) is compact and disjoint from \(C\), by [eq:coordinate-injectivity]. Its complement is therefore a relative open neighborhood of \(C\) in \(W\). Subdivide \(W\) into sufficiently small closed cubes and take the union \(P\) of all cubes meeting \(C\). Finiteness and the positive distance from \(C\) to the bad projection ensure that \(P\) is contained in this neighborhood. If the bad projection is empty, any such subdivision suffices. This construction is relative to \(W\); it does not extend the first \(M\) coordinates beyond the unit cube.

Regard \(j_0\) as a map \(j:Y\to P\) and put \(f_P=f_0|_P\). The map \(f_P\) is coordinate-linear and hence PL. On \(\mathcal T(f_P,Z)\), the restriction of \(B(u,z)\) to the slot bundle is bounded below by \(2\delta\) on unit vectors, and thus is the required continuous bundle embedding. Finally \(f_P\circ j=f\) by the choice of the first \(M\) coordinates. ◻

Proof of 3. Apply 20 to obtain \(j:Y\to P\) and \(f_P\) with the same bundle embedding and \(f=f_P\circ j\). By 19 the \(MU\) point test of \(f_P\) vanishes. 17 gives the stated compression maps on \(P^u\) for arbitrarily large \(u\). Composing with \(j^{\times u}\) gives the maps on \(Y^u\). The factorization of \(f\) is exact, so the boundary agreement also pulls back exactly. ◻