Addendum 2.5 Starting with $C_{0,q}\leqslant \widetilde C_{0,q}=\pi _0(F_q,\,\zeta _q)\cong \pi _1(BL_q,\,l_q(\zeta _{q-1}))$ with incoming differential $\tilde d\colon C_{1,q-1}\xrightarrow {d}C_{0,q}\leqslant \widetilde C_{0,q}$, each derived group $C_{0,q}^{(n)}$ is a subgroup of the group $\widetilde C_{0,q}^{(n)}=\widetilde C_{0,q}^{(n-1)}/\mathop {\rm im}\nolimits \tilde d^{(n-1)}$ with incoming differential $\tilde d^{(n)}\colon C^{(n)}_{1,q-n-1}\xrightarrow {d^{(n)}}C^{(n)}_{0,q}\leqslant \widetilde C_{0,q}^{(n)}$. Moreover, $\widetilde C^{(n)}_{0,q}$ and $\tilde d^{(n)}$ are natural with respect to morphisms of towers induced by maps of classifying spaces $BL_q$. The exact couple $(A,\,C)$ together with natural subgroup inclusions $C^{(n)}_{0,q}\leqslant \widetilde C^{(n)}_{0,q}$ will be called an augmented Federer's exact couple and denoted $(A,\,C,\,\widetilde C)$.

Proof. As $Z_q\to Z_{q-1}\xrightarrow {l_q}BL_q$ is a homotopy fibration, $\partial \colon \pi _{p+1}(Z_{q-1},\,\zeta _{q-1})\to \pi _{p}(F_q',\,\zeta _q)=C_{p,q}$ can be viewed as $(l_q)_*\colon \pi _{p+1}(Z_{q-1},\,\zeta _{q-1})\to \pi _{p+1}(BL_q,\,l_q(\zeta _{q-1}))$. Moreover, $C_{0,q}$ can be identified with the image of $(l_q)_*$ in $\pi _1(BL_q,\,l_q(\zeta _{q-1}))$ which is abelian by our assumption on $BL_q$. This provides the identification of $C_{p,q}$ and that of $\partial$ in the differential $\partial \circ i\colon C_{p,q}\to C_{p-1,q+1}$. More precisely, as the fibration $Z_q\to Z_{q-1}$ is principal, $F_q$ is homotopy equivalent to the loop-space $\Omega (BL_q,\,l_q(\zeta _{q-1}))$, and the connecting morphism in the homotopy exact sequence of $EL_q\to BL_q$ induces the isomorphism $\delta _q\colon \pi _{p+1}(BL_q,\,l_q(\zeta _{q-1}))\to \pi _p(F_q,\,\zeta _q)$. Noting that $\partial =(\delta _q)^{-1}\circ (l_q)_*\colon \pi _{p+1}(Z_{q-1},\,\zeta _{q-1})\to \pi _p(F_q,\,\zeta _q)$ completes the proof of the main statement.

For remark 2.3, consider a general fibration $f\colon E\to B$ and let $f(e_0)=b_0$ be any coherent pair of basepoints. The connecting morphism $\pi _*(\Omega B,\,\mathop {\rm const}\nolimits _{b_0})\to \pi _{*}(F,\,e_0)$ is induced by a map $\Omega (B,\,b_0)\to F$ which is most easily constructed by means of a lifting function that continuously lifts loops at $b_0$ to paths in $E$ beginning at $e_0$ and takes the end point (in $F$). A regular fibration admits a lifting function sending constant paths to constant paths, forcing $\mathop {\rm const}\nolimits _{b_0}$ to map to $e_0$.

To prove the addendum by induction, note that if $\tilde d^{(n-1)}\colon C^{(n-1)}_{1,q-n}\to \widetilde C^{(n-1)}_{0,q}$ is just $d^{(n-1)}$ followed by the inclusion $C^{(n-1)}_{0,q}\leqslant \widetilde C^{(n-1)}_{0,q}$, the images of $\tilde d^{(n-1)}$ and $d^{(n-1)}$ in $\widetilde C^{(n-1)}_{0,q}$ coincide, yielding subgroup inclusion $C^{(n)}_{(0,q)}=C^{(n-1)}_{0,q}/\mathop {\rm im}\nolimits d^{(n-1)}\leqslant \widetilde C^{(n-1)}_{0,q}/\mathop {\rm im}\nolimits \tilde d^{(n-1)}=\widetilde C^{(n)}_{(0,q)}$.

Proof. There are induced pullback diagrams as follows. (Amend [Reference Smrekar29, Lemma A.3].)

(3.3)

Here, $\psi _q$ denotes the composite $A\xrightarrow {\phi _q}Y_q\to P(BK_q)$. Also, let $\Psi _q$ denote the composite $X\xrightarrow {\zeta _q}Y_q\to P(BK_q)$. Note that $\varepsilon \psi _q=k_q\phi _{q-1}$ and $\varepsilon \Psi _q=k_q\zeta _{q-1}$. As mentioned above, the vertical arrows in (3.3) are fibrations. As $P(BK_q)$ is contractible, $\mathop {\rm map}\nolimits (X,\,P(BK_q))\to \mathop {\rm map}\nolimits (A,\,P(BK_q))$ is a fibration and homotopy equivalence. Thus $EL_q=\mathop {\rm map}\nolimits _{\psi _q}(X,\,P(BK_q))$, its fibre over $\psi _q$, is also contractible, rendering $(\eta _q)_*$ in diagram (3.3) a principal fibration. Set $BL_q=\mathop {\rm map}\nolimits _{\varepsilon \psi _q}(X,\,BK_q)$. Assuming an H-group multiplication on $BK_q$ with strict unit $\mathbb {1}$, we get induced H-group structures on $\mathop {\rm map}\nolimits (X,\,BK_q)$ and $\mathop {\rm map}\nolimits (A,\,BK_q)$ and, therefore, an induced map of fibrations as follows.

(3.4)

The horizontal arrows are homotopy equivalences and by coglueing homotopy equivalences (see [Reference Brown and Heath4, Corollary 1.5] or [Reference May and Ponto19, Lemma 2.2.4]), so are the maps on the fibres. In particular,

(3.5)\begin{equation} \mathop{\rm map}\nolimits_{A\to\mathbb{1}}(X,BK_q)\to\mathop{\rm map}\nolimits_{\varepsilon\psi_q}(X,BK_q) \end{equation}

is a homotopy equivalence. This shows that, in fact, $BL_q$ is an H-group and so lemma 2.1 can be applied. The identification of $C_{p,q}$ as (a subgroup of) $\check H^{n_q-p}(X,\,A;\pi ^q)$ comes via (3.5) through the obvious homeomorphism of loop spaces

\[ \Omega^k\big(\mathop{\rm map}\nolimits_{A\to\mathbb{1}}(X,BK_q),\mathop{\rm const}\nolimits_{\mathbb{1}}\!\big)\approx\mathop{\rm map}\nolimits\big((X,A),(\Omega^k(BK_q,\mathbb{1}),\mathop{\rm const}\nolimits_{\mathbb{1}}\!\big) \]

and representability of Čech cohomology over paracompact Hausdorff pairs (see [Reference Huber14] and [Reference Goto11]). The reader may check that this identification is natural with respect to maps of pairs $(X,\,A)\to (X',\,A')$ over $Y\!$.

For full naturality with respect to maps of towers (3.1), we assume that

(3.6)\begin{equation} K(\pi^q_{(1)},n_q+1)\to K(\pi^q_{(2)},n_q+1) \end{equation}

are strict H-maps of ANR H-groups with strict units. To achieve that, we deloop a given homotopy representative for (3.6) to a based map $K(\pi ^q_{(1)},\,n_q+2)\to K(\pi ^q_{(2)},\,n_q+2)$ between ANRs and take the induced map between the associated Moore loop spaces (which will also be ANRs).

Addendum 3.2 The first differential $C_{p,q}\to C_{p-1,q+1}$ for $p\geqslant 1$ is obtained by applying $\pi _p$ to the composite of (based) continuous maps

(3.7)\begin{equation} \begin{aligned} \Omega\big(\mathop{\rm map}\nolimits_{A\to\mathbb{1}}(X,BK_q),\mathop{\rm const}\nolimits_{\mathbb{1}}\!\big)\to\Omega\big(\mathop{\rm map}\nolimits_{k_q\phi_{q-1}}(X,BK_q),k_q\zeta_{q-1}\big)\to\\ \left\{f\,\vert\,\eta_q f=\zeta_{q-1}\right\}\subset\mathop{\rm map}\nolimits_{\phi_q}(X,Y_q)\xrightarrow{(k_{q+1})_*}\mathop{\rm map}\nolimits_{k_{q+1}\phi_q}(X,BK_{q+1}) \end{aligned} \end{equation}

The first map is induced by the self-homotopy equivalence $\mathop {\rm map}\nolimits _{A\to \mathbb {1}}(X,\,BK_q)\to \mathop {\rm map}\nolimits _{k_q\phi _{q-1}}(X,\,BK_q)$ sending the map $\mathop {\rm const}\nolimits _{\mathbb {1}}$ to $k_q\zeta _{q-1}$ that is in turn induced by an H-group multiplication on $BK_q$ with strict unit $\mathbb {1}$. The following 3 maps correspond to the sequence (2.4) in conjunction with remark 2.3. This identification of the differential is natural in $(X,\,A)$ and $Y\!$.

Proof. The dimension assumption guarantees a cofinal subfamily $\{\lambda \}$ with $\mathop {\rm dim}\nolimits N_\lambda \leqslant d$. Let $N_\lambda$ be such and let $T$ be a compact $k$-dimensional polyhedron. Consider

(4.1)\begin{equation} [N_\lambda\times T,Y]\to[N_\lambda\times T,P^nY]. \end{equation}

By relative cellular approximation, (4.1) is a surjection if $k+d\leqslant n+1$ and an injection if $k+d\leqslant n$. By theorem 4.1, $[X\times T,\,Y]\to [X\times T,\,P^nY]$ has the same properties. In particular, taking $T=\left \{*\right \}$ we infer that $[X,\,Y]\to [X,\,P^nY]$ is a bijection. Let $f\colon X\to Y$ be a basepoint in $\mathop {\rm map}\nolimits (X,\,Y)$ and let $*$ denote a basepoint in the sphere $S^k$. Making use of the projection retraction $X\times S^k\to X\times \left \{*\right \}\equiv X$ as in the proof of [Reference Klein, Schochet and Smith16, Theorem 6.4], we infer that

(4.2)\begin{equation} [X\times S^k,Y]_{f}\to[X\times S^k,Y]\to[X,Y] \end{equation}

is a split short exact sequence of pointed sets. Here, $[X\times S^k,\,Y]_{f}$ is the set of homotopy classes of maps $X\times S^k\to Y$ that restrict to $f\colon X\times \left \{*\right \}\to Y$. Applying naturality of (4.2) to $p\colon Y\to P^nY$, a diagram chase shows that $p_*\colon [X\times S^k,\,Y]_f\to [X\times S^k,\,P^nY]_{pf}$, which can clearly be identified with $p_*\colon \pi _k(\mathop {\rm map}\nolimits (X,\,Y),\,f)\to \pi _k(\mathop {\rm map}\nolimits (X,\,P^nY),\,pf)$, is bijective for $k< n-d+1$ and surjective for $k=(n-d+1)$.

For $p_*\colon \mathop {\rm map}\nolimits _{A\to y_0}(X,\,Y)\to \mathop {\rm map}\nolimits _{A\to p(y_0)}(X,\,P^nY)$, use pairs everywhere.

Proof. Case (i) follows from proposition 4.2 applied to the morphism of the homotopy exact sequences of fibrations $\mathop {\rm map}\nolimits (X,\,Y)\to \mathop {\rm map}\nolimits (A,\,Y)$ and $\mathop {\rm map}\nolimits (X,\,P^nY)\to \mathop {\rm map}\nolimits (A,\,P^nY)$. For (ii), consider the pullback diagrams induced by (1.2):

(4.4)

Using proposition 6.1, it follows that the vertical arrows in (4.4) are fibrations. Reasoning as in case (i), it follows that the mapping induced on the fibres of

is an $(n-d)$-equivalence. By naturality of (4.4) in $Y\!$,

\[ \mathop{\rm map}\nolimits_\phi(X_{i},Y)\to\mathop{\rm map}\nolimits_{p\phi}(X_{i},P^nY) \]

is a weak $(n-d)$-equivalence for all $i$. Now $\mathop {\rm map}\nolimits _\phi (X,\,Y)$ is the inverse limit of $\left \{\mathop {\rm map}\nolimits _\phi (X_{i},\,Y)\right \}$ (see corollary 6.2) and there are the associated $\lim ^1--\lim$ short exact sequences (see [Reference Bousfield and Kan5, Chapter IX] and also [Reference Mardešić and Segal17, p. 178, Theorem 1]). By naturality in $Y$, we get morphisms of those exact sequences identifying the morphisms $\pi _k(\mathop {\rm map}\nolimits _\phi (X,\,Y),\,f)\to \pi _k(\mathop {\rm map}\nolimits _\phi (X,\,P^nY),\,pf)$. From those we infer (ii).

Proof of theorem 1.1 Assume the notation of theorem 3.1. Diagram (3.3) exhibits $\mathop {\rm map}\nolimits _{\phi _q}(X,\,Y_q)$ as the homotopy fibre of $(k_q)_*\colon \mathop {\rm map}\nolimits _{\phi _{q-1}}(X,\,Y_{q-1})\to BL_q$ where $BL_q$ is an H-group and hence nilpotent. As $\mathop {\rm map}\nolimits _{\phi _{0}}(X,\,Y_{0})=\left \{*\right \}$, it follows by induction that $\mathop {\rm map}\nolimits _{\phi _q}(X,\,Y_q)$ are nilpotent for all $q$ (see [Reference Hilton, Mislin and Roitberg12, Theorem II.2.2] together with the final remark of the proof, as well as the proof of [Reference Hilton, Mislin, Roitberg and Steiner13, Theorem A]).

Case 1. $Y$ has finitely many nontrivial homotopy groups. Then $Y\to Y_q$ is a homotopy equivalence for some $q$. This finishes the proof if $A=\emptyset$ or if $\phi$ maps the entire $A$ to a nondegenerate $y_0\in Y$, for then we have homotopy equivalences $\mathop {\rm map}\nolimits (X,\,Y)\to \mathop {\rm map}\nolimits (X,\,Y_q)$ and $\mathop {\rm map}\nolimits _{A\to y_0}(X,\,Y)\to \mathop {\rm map}\nolimits _{A\to \eta ^q(y_0)}(X,\,Y_q)$. (See the proof of (1) in the discussion on page 7.)

Suppose that $A\hookrightarrow X$ is a cofibration or $(X,\,A)$ is a proper pair and $Y$ is an ANR. Then we have restriction fibrations $\mathop {\rm map}\nolimits (X,\,Y)\to \mathop {\rm map}\nolimits (A,\,Y)$ and $\mathop {\rm map}\nolimits (X,\,Y_q)\to \mathop {\rm map}\nolimits (A,\,Y_q)$ (note $Y_q$ is an ANR by construction), and a morphism between them induced by the homotopy equivalence $Y\to Y_q$. By [Reference Brown and Heath4, Corollary 1.5], we get the homotopy equivalence $\mathop {\rm map}\nolimits _\phi (X,\,Y)\to \mathop {\rm map}\nolimits _{\phi _q}(X,\,Y)$ between the fibres. Thus in Case 1, the assumptions of proposition 2.6 and corollary 2.7 are met trivially.

Case 2. $Y$ has infinitely many nontrivial homotopy groups. By proposition 4.2 and corollary 4.3, the connectivity of $\mathop {\rm map}\nolimits _\phi (X,\,Y)\to \mathop {\rm map}\nolimits _{\phi _q}(X,\,Y_q)$ tends to infinity with $q$, implying nilpotency of $\mathop {\rm map}\nolimits _\phi (X,\,Y)$ as well as the assumptions of proposition 2.6 and corollary 2.7.

Finally, we set $E^2_{-k,q}=C_{q-k,q}$ for $k< q$ and $E^2_{-q,q}=\widetilde C_{0,q}$ to obtain the upper half-plane spectral sequence with homology-type differentials as claimed. Localization on the level of $E^2$ implies localization in the limit in case of convergence by theorem 2.8.

Proof. As $R$ and $R\vert _{\Gamma (X)}$ are maps between metrizable spaces, it suffices to prove the homotopy lifting property for metrizable spaces (see [Reference Dugundji9, XX.2.3]).

To prove (i), we take a metric space $Z$. The solid part of the lifting diagram

(6.2)

induces a map $\eta \colon Z\times X\times \left \{0\right \}\cup Z\times A\times [0,\,1]\to Y$. Being the product of a paracompact and a compact space, $Z\times X$ is paracompact (Barratt [Reference Barratt1, Lemma 12.43]) and hence collection-wise normal. Therefore its $P_0$-embedded subsets are precisely its zero sets ([Reference Stramaccia34, Examples 1.2]). Thus, $Z\times A$ is $P_0$-embedded in $Z\times X$, and Morita's extension theorem (see [Reference Stramaccia34, Theorem 2.2]) yields a map $Z\times X\times [0,\,1]\to Y$ extending $\eta$. Its adjoint is the lifting sought in (6.2).

We turn to (ii). The lifting problem analogous to (6.2) but with $\mathop {\rm map}\nolimits (X,\,Y)$ and $\mathop {\rm map}\nolimits (A,\,Y)$ replaced by, respectively, $\Gamma (X)$ and $\Gamma (A)$, is equivalent to

(6.3)

By the proof of (i), $\eta$ extends to a map $\bar \eta \colon Z\times X\times [0,\,1]\to Y$. We remedy the possible difference between $p\circ \bar \eta$ and $f\circ \mathop {\rm pr}\nolimits _X$ (where $\mathop {\rm pr}\nolimits _X$ is projection onto $X$) as follows. Let a partial map into $B$ be defined by $p\circ \bar \eta$ on $Z\times X\times [0,\,1]\times \left \{0\right \}$ and $f\circ \mathop {\rm pr}\nolimits _X$ on $Z\times X\times [0,\,1]\times \left \{1\right \}\cup (Z\times X\times \left \{0\right \}\sqcup Z\times A\times [0,\,1])\times [0,\,1]$. This partial map is defined on a zero (hence $P_0$-embedded) set in $Z\times X\times [0,\,1]\times [0,\,1]$ and since $B$ is an AR, there exists an extension over all $Z\times X\times [0,\,1]\times [0,\,1]$ (see [Reference Stramaccia34, Theorem 2.1]). We consider the lifting problem

By construction, the homotopy into $B$ is stationary on $Z\times X\times \left \{0\right \}\cup Z\times A\times [0,\,1]$. By regularity of $p$ (see [Reference Dugundji9, XX.2.4]), the lifting into $Y$ can also be assumed to be stationary there. Level 1 of that lifting is a map $Z\times X\times [0,\,1]\to Y$ that extends $\eta$ and projects to $f\circ \mathop {\rm pr}\nolimits _X$ by $p$, and thus it constitutes the lifting sought in (6.3).

Proof. The canonical inverse limit is the subspace $Z'\subset \prod _{i=1}^\infty \mathop {\rm map}\nolimits _\phi (X_i,\,Y)$ consisting of compatible sequences of maps $f_i\colon X_i\to Y$ with $f_i\vert _A=\phi$. Restrictions $R^i$ furnish the obvious homeomorphism $R\colon \mathop {\rm map}\nolimits _\phi (X,\,Y)\to Z'$; it is an embedding since the $X_i$ dominate compact sets in $X$ and onto because $X=\mathop {\rm colim}\nolimits _iX_i$. In diagram (4.4), restriction $\mathop {\rm map}\nolimits (C_i,\,Y)\to \mathop {\rm map}\nolimits (S_i,\,Y)$ is a fibration by (i) of proposition 6.1, and hence so is the pullback restriction $\mathop {\rm map}\nolimits _\phi (X_i,\,Y)\to \mathop {\rm map}\nolimits _\phi (X_{i-1},\,Y)$. The spaces $\mathop {\rm map}\nolimits (C_i,\,Y)$ and $\mathop {\rm map}\nolimits (S_i,\,Y)$ have CW homotopy type by [Reference Milnor22, Theorem 3]. Now $\mathop {\rm map}\nolimits _\phi (X_0,\,Y)$ is a point and it follows from [Reference Smrekar29, Proposition 4.2] that all $\mathop {\rm map}\nolimits _\phi (X_i,\,Y)$ have CW homotopy type.

By way of composition, $\mathop {\rm map}\nolimits _\phi (X_j,\,Y)\to \mathop {\rm map}\nolimits _\phi (X_i,\,Y)$ are fibrations for all $j\geqslant i$. The universal property of the limit implies that also the $R^i$ are fibrations.

Proof of theorem 1.6 Corollary 1.3 guarantees that the maps

(6.4)\begin{equation} \mathop{\rm map}\nolimits_\phi(X_i,Y)\to\mathop{\rm map}\nolimits_{l\phi}(X_i,Y_{(P)}) \end{equation}

localize homotopy groups. Corollary 6.2 identifies the inverse limit of maps (6.4) as $\mathop {\rm map}\nolimits _\phi (X,\,Y)\to \mathop {\rm map}\nolimits _{l\phi }(X,\,Y_{(P)})$. By [Reference Smrekar31, Theorem 6.1], the spaces $\mathop {\rm map}\nolimits (X,\,Y)$ and $\mathop {\rm map}\nolimits (X,\,Y_{(P)})$ are nilpotent and this latter map localizes homotopy groups.

Proof of proposition 1.8 Assume the notation of § 3. Diagram (3.3) exhibits $Z_q=\mathop {\rm map}\nolimits _{\phi _q}(X,\,Y_q)$ as the homotopy fibre of $(k_q)_*\colon Z_{q-1}\to \mathop {\rm map}\nolimits _{A\to *}(X,\,BK_q)$ where $BK_q$ is an Eilenberg–MacLane space. Now $\mathop {\rm map}\nolimits _{A\to *}(X,\,BK_q)$ has CW type since $(X,\,A)$ is quasi-finite, and as $Z_0=\left \{*\right \}$, an inductive application of [Reference Stasheff33, Proposition 0] shows that all $Z_q$ have CW homotopy type.

Set $Z^i=\mathop {\rm map}\nolimits _\phi (X_i,\,Y)$, $\zeta ^i=\zeta \vert _{X_i}$, $Z^i_q=\mathop {\rm map}\nolimits _\phi (X_i,\,Y_q)$, and $\zeta ^i_q=\zeta _q\vert _{X_i}$. By corollary 6.2, $Z=\mathop {\rm map}\nolimits _\phi (X,\,Y)$ is the limit of $Z^i$ and $Z_q$ is the limit of $Z^i_q$ for all $q$, and all $Z^i$ and $Z^i_q$ have CW homotopy type. To establish the (additional) assumption of corollary 2.7, we show that for every $p\geqslant 1$, restrictions $\pi _p(Z,\,\zeta )\to \pi _p(Z_q,\,\zeta _q)$ are injective for all big enough $q$. By [Reference Smrekar29, Corollary 3.4,(i)], applied to $Z=\lim _iZ^i$, there exists an $i$ such that $\pi _p(Z,\,\zeta )\to \pi _p(Z^i,\,\zeta ^i)$ are injective for all $p\geqslant 1$. Fix some $p\geqslant 1$. By corollary 4.3, $\pi _p(Z^i,\,\zeta ^i)\to \pi _p(Z^i_q,\,\zeta ^i_q)$ is an isomorphism for all big enough $q$. Thus, $\pi _p(Z,\,\zeta )\to \pi _p(Z_q,\,\zeta _q)$ must be injective for those $q$.

To establish the assumptions of proposition 2.6, fix some $q$. By another application of [Reference Smrekar29, Corollary 3.4,(i)], this time to $Z_q=\lim _iZ^i_q$, there exists an $i$ such that $\pi _p(Z_q,\,\zeta _q)\to \pi _p(Z^i_q,\,\zeta ^i_q)$ are injective for all $p\geqslant 1$. By [Reference Smrekar29, Corollary 3.4,(ii)], applied to $Z=\lim _iZ^i$, there exists a $j\geqslant i$ such that the image of $\pi _p(Z,\,\zeta )\to \pi _p(Z^i,\,\zeta ^i)$ coincides with that of $\pi _p(Z^j,\,\zeta ^j)\to \pi _p(Z^i,\,\zeta ^i)$ for all $p$. Finally, we fix some $p\geqslant 1$. By corollary 4.3, there exists a $t$ such that $\pi _p(Z^j,\,\zeta ^j)\to \pi _p(Z^j_t,\,\zeta ^j_t)$ is an isomorphism. A straightforward diagram chase shows that every element in the image of $\pi _p(Z_t,\,\zeta _t)\to \pi _p(Z_q,\,\zeta _q)$ is also in the image of $\pi _p(Z,\,\zeta )\to \pi _p(Z_q,\,\zeta _q)$.

Proof of proposition 1.9 By homotopy invariance of $\mathop {\rm map}\nolimits _{A\to *}(X,\,Y)$ for a well-pointed $Y$, we may assume that $Y=K(G,\,n)$ is an H-group with strict unit $\mathbb {1}$ and an ANR. If $\left \{X_i\right \}$ is any admissible chain for $(X,\,A)$, then by corollary 6.2, $Z^i=\mathop {\rm map}\nolimits _{A\to \mathbb {1}}(X_i,\,Y)$ defines a tower of fibrations whose terms have CW homotopy type and whose limit is $Z=\mathop {\rm map}\nolimits _{A\to \mathbb {1}}(X,\,Y)$. Homotopical representation of Čech cohomology directly implies that restriction $Z\to Z^i$ induces the cohomological restriction $\check H^n(X,\,A;G)\to \check H^n(X_i,\,A;G)$.

Let $\zeta \colon X\to Y$ be a map with restrictions $\zeta ^i=\zeta \vert _{X_i}$. Pointwise multiplications by $\zeta$, respectively $\zeta ^i$, comprise a homotopy auto-equivalence of the tower $\left \{Z^i\right \}$ and its limit that maps the components of $\mathop {\rm const}\nolimits _{\mathbb {1}}$ to those of $\zeta$, respectively $\zeta ^i$. Thus we can identify the morphisms $\pi _k(Z^j,\,\zeta ^j)\to \pi _k(Z^i,\,\zeta ^i)$ with $\check H^{n-k}(X_j,\,A;G)\to \check H^{n-k}(X_i,\,A;G)$. An application of [Reference Smrekar31, Theorem 4.1] concludes the proof.

Proof. (i) By the assumption on $B$ and normality of the $C_i$ in (1.2), we may refine each step $X_{i-1}\subset X_i$ into a finite chain of adjunctions to assume, after reindexing, that $f(\gamma _i(C_i))\subset K_i$ for a contractible open set $K_i$ in $B$. We get induced pullbacks

(7.1)

By proposition 6.1, the vertical arrow on the right is a fibration, and hence so is the one on the left. Consider $p_*\colon \mathop {\rm map}\nolimits (C_i,\,E\vert _{K_i})\to \mathop {\rm map}\nolimits (C_i,\,K_i)$. As $C_i$ is compact, this is a fibration between spaces of CW homotopy type. In fact $\mathop {\rm map}\nolimits (C_i,\,K_i)$ is contractible and hence fibre inclusion $\Gamma (C_i,\,E\vert _{K_i})\hookrightarrow \mathop {\rm map}\nolimits (C_i,\,E\vert _{K_i})$ is a homotopy equivalence. On the other hand, $Y\hookrightarrow E\vert _{K_i}$ is also a homotopy equivalence, hence $\mathop {\rm map}\nolimits (C_i,\,Y)\simeq \Gamma (C_i,\,E\vert _{K_i})$ is nilpotent (under the additional hypothesis) by theorem 1.1. The same holds for $S_i$ in place of $C_i$ and since $\Gamma _\phi (X_0,\,E)$ is a one point-space, it follows by induction that all $\Gamma _\phi (X_{i},\,E)$ are nilpotent (see May and Ponto [Reference May and Ponto19, Proposition 4.4.3]) of CW type (see Stasheff [Reference Stasheff33, Propositions 0 and 12]).

(ii) We apply the same reasoning to $\Gamma _{l\phi }(X_i,\,E_{(P)})$ and note that $\Gamma (C_i,\,E\vert _{K_i})\to \Gamma (C_i,\,E_{(P)}\vert _{K_i})$ and $\Gamma (S_i,\,E\vert _{K_i})\to \Gamma (S_i,\,E_{(P)}\vert _{K_i})$ may be identified with, respectively, $\mathop {\rm map}\nolimits (C_i,\,Y)\to \mathop {\rm map}\nolimits (C_i,\,Y_{(P)})$ and $\mathop {\rm map}\nolimits (S_i,\,Y)\to \mathop {\rm map}\nolimits (S_i,\,Y_{(P)})$ that localize by corollary 1.3. Thus if $\Gamma _\phi (X_{i-1},\,E)\to \Gamma _\phi (X_{i-1},\,E_{(P)})$ localizes (on path components), then so does $\Gamma _\phi (X_{i},\,E)\to \Gamma _\phi (X_{i},\,E_{(P)})$ by [Reference May and Ponto19, Proposition 6.2.5].

Proof of theorem 1.10 The theorem deals with absolute sections ($A=\emptyset$), and we may take a representative for $p$ as specified prior to the statement of theorem 7.2. Thus, statement (a) follows directly from theorem 7.2 as then $X=X_i$ for some $i$.

For (b), note that $\Gamma (X,\,E)\to \Gamma (X,\,E_{(P)})$ is the limit of maps $\Gamma (X_i,\,E)\to \Gamma (X_i,\,E_{(P)})$ (which localize by theorem 7.2). Apply [Reference Smrekar31, Theorems 6.1 and 6.2].