Based loop spaces of CW complexes have CW homotopy type

We first extend [F1] to arbitrary relative CW source pairs under AC, retaining the exact prescribed boundary homotopy. For one relative disk $(D^d,S^{d-1})$, [F1] supplies $w$ and $J$ with boundary $T(z,\lambda (t))$, where $\lambda (t)=\min (4t,1)$. On $B=(S^{d-1}\times I)\cup (D^d\times \{0,1\})\subset D^d\times I$ change this boundary homotopy through $T(z,(1-s)\lambda (t)+st)$, leaving the two end-disk maps unchanged. These prescriptions agree at their intersections. The finite CW inclusion $B\subset D^d\times I$ has HEP by [F2]; applying it with initial map $J$ produces, at $s=1$, a homotopy with the same end maps and boundary exactly $T(z,t)$. For a zero-cell [F1] simply chooses a source point and a path from its image to the prescribed target point, with the direction reversed if necessary. Now extend a given source map and target homotopy over all cells of a relative CW pair, in increasing dimension, using this corrected disk assertion on each cell. AC selects all disk witnesses at each stage; recursion on dimensions gives maps and homotopies agreeing literally on the preceding stage and on the prescribed subcomplex. The CW quotient test and its product with $I$ in [F2] prove continuity of their unions. Thus for any weak equivalence $f:P\to Y$ and any CW pair $(K,L)$, a source map on $L$, a target map on $K$ and an exact boundary homotopy extend to a source map on $K$ and a target homotopy with exactly that boundary homotopy.

We next construct a simplicial homotopy model of an arbitrary CW complex, with no finiteness hypothesis. Here is the finite building block, with its target triangulation retained. For a simplicial map $f:P\to Q$ of finite complexes, construct a simplicial complex $M(f)$ inductively over source simplices. Start with the target $Q$. For each source simplex $\sigma$ with image face $\tau =f(\sigma )$, the already constructed boundary part $M(f{\mathrm{\mid }}_{\mathrm{\partial }\sigma }:\mathrm{\partial }\sigma \to \tau )$ includes $\tau$; adjoin the cone on that boundary part with a fresh apex $b_{\sigma }$. Use the same previously constructed pieces on common faces. The cones on the subdivided source boundaries form a copy of $\mathrm{sd}P$, disjoint from $Q$. Each boundary part strongly deformation retracts onto $\tau$, by induction, so both it and its cone are contractible. Its inclusion in its cone is a finite CW homotopy equivalence, and [F5] gives a strong deformation retraction of the cone onto its base. Apply these retractions in decreasing source simplex dimension, simultaneously on pieces of the same dimension; each fixes the shared boundary pieces. The result strongly deformation retracts $M(f)$ onto $Q$. Each source simplex stays in its image face under the endpoint retraction $r$, so $r{\mathrm{\mid }}_{\mathrm{sd}P}$ is homotopic to $f$ there by the straight-line homotopy in that face. Use finite HEP [F2] on the disjoint subcomplexes $Q$ and $\mathrm{sd}P$ to extend this endpoint adjustment over $M(f)$, keeping $Q$ fixed. Concatenating with the first retraction gives a strong deformation from ${\mathrm{id}}_{M(f)}$ to a retraction whose restriction to the source is exactly $f$. In particular the source inclusion is homotopic in $M(f)$ to the map $f$ into $Q$. Finally $M(f){\cup }_{\mathrm{sd}P}C(\mathrm{sd}P)$ is simplicial. Fresh apex and source vertices and the common-face construction make every intersection an actual subcomplex.

We begin a general metric-space domination argument, independent of the later loop-space construction. Let $Z$ be a metric space, let $U_Z$ be an open neighborhood of its diagonal in $Z\times Z$, and let ${\mathrm{\Lambda }}_Z:U_Z\times I\to Z$ be continuous with ${\mathrm{\Lambda }}_Z(a,b,0)=a$, ${\mathrm{\Lambda }}_Z(a,b,1)=b$ and ${\mathrm{\Lambda }}_Z(a,a,t)=a$. Suppose $\mathcal{V}$ is an open cover such that $V\times V\subset U_Z$ and ${\mathrm{\Lambda }}_Z(V\times V\times I)\subset V$ for each $V\in \mathcal{V}$; these are the convex sets for this argument. We will construct a CW domination in this general setting. Use any bounded compatible metric $d$. Define the positive 1-Lipschitz function $\rho (z)={\sup }_{V\in \mathcal{V}}\min (1,d(z,Z\setminus V))$, where the contribution for $V=Z$ is defined to be $1$. The cover $\mathcal{B}=\{B(z,\rho (z)\mathrm{/}8):z\in Z\}$ has the property that the union of its members containing any fixed $a$ lies in a convex set: if $a\in B(z,\rho (z)\mathrm{/}8)$, the Lipschitz inequality gives $\rho (z)<8\rho (a)\mathrm{/}7$, so this ball lies in $B(a,2\rho (a)\mathrm{/}7)$. Some $V\in \mathcal{V}$ has $d(a,Z\setminus V)>\rho (a)\mathrm{/}2$ (or is $Z$), by the definition of the supremum, and contains that larger ball. Take a locally finite open refinement $\mathcal{W}$ of $\mathcal{B}$ by [F7]. For every $a$, the union of all $W\in \mathcal{W}$ containing $a$ still lies in one such convex $V$. Discard empty members. This supplies the precise refinement property needed below without invoking an unproved star-refinement theorem.

Separately, based loop homotopy type is preserved by an unbased homotopy equivalence between connected spaces, without any cofibration hypothesis on their basepoints. Here are the details needed for the possibly nonvertex basepoint $x$. Given $f:X\to Y$, inverse $g$, homotopies $H:gf\simeq {\mathrm{id}}_X$ and $T:fg\simeq {\mathrm{id}}_Y$, chosen basepoints $x,y$, and a path $a:y\to f(x)$, define $L(\gamma )=a\ast (f\gamma )\ast a^{-1}$. Put $b=H_x^{-1}\ast (ga{)}^{-1}:x\to g(y)$ and $G(\delta )=b\ast (g\delta )\ast b^{-1}$. Use fixed lengths for finite path concatenations; reparametrizing between different parenthesizations gives continuous homotopies. The square $H(\gamma (t),s)$ shows that $H_x^{-1}\ast (gf\gamma )\ast H_x$ is homotopic through loops to $\gamma$; insert or delete the backtracking paths $ga\ast (ga{)}^{-1}$ to conclude $GL\simeq \mathrm{id}$. The square $T(\delta (t),s)$ similarly gives $LG\simeq C_c$, where $C_c(\delta )=c\ast \delta \ast c^{-1}$ and $c=a\ast (fb)\ast T_y$ is a fixed loop at $y$. Backtracking cancellation makes $C_c$ a homotopy equivalence with inverse $C_{c^{-1}}$. Consequently $R=G{C}_{c^{-1}}$ is a right homotopy inverse of $L$, while $G$ is a left homotopy inverse; the identities $G\simeq GLR\simeq R$ show that either is a two-sided homotopy inverse. For completeness, a path followed by its reverse contracts by replacing its triangular traversal parameter $\min (2t,2-2t)$ by $(1-s)\min (2t,2-2t)$; this depends continuously on the path. The displayed squares, this formula and the interval exponential law [F2] make all these homotopies continuous in the variable loop.

By step 1.1, postcomposition by a weak equivalence $f:P\to Y$ induces a bijection $[K,P]\to [K,Y]$ on unbased homotopy classes for every CW complex $K$. Surjectivity is step 1.1 with empty fixed subcomplex. For injectivity, given a homotopy $K\times I\to Y$ between $f{a}_0$ and $f{a}_1$, use step 1.1 on the CW pair $(K\times I,K\times \{0,1\})$, the prescribed end maps $a_0,a_1$ and the constant end homotopies. The resulting map $K\times I\to P$ is a homotopy between those end maps. The interval has its finite CW structure, so the cylinder pair is a CW pair; equivalently its cell prisms attach by the characteristic-disk quotient in [F2].

Starting with $K_0=X^0$ and a common space $Z_0=X^0$, construct simplicial complexes $K_d$ and common spaces $Z_d$ containing both $X^d$ and $\mathrm{\mid }{K}_d\mathrm{\mid }$, with strong deformation retractions onto each, whose endpoint retractions restrict to the preceding endpoint retractions on $Z_{d-1}$. Suppose these data are available through $d-1$. For every $d$-cell of $X$, compose its attaching map $a_e:S^{d-1}\to X^{d-1}\subset Z_{d-1}$ with the endpoint retraction to $K_{d-1}$. Its compact image lies in a finite subcomplex by [F4]. Give the sphere a finite triangulation and approximate this map by a simplicial $f_e:P_e\to Q_e\subset K_{d-1}$ after a finite subdivision, using [F4]. AC selects this finite data for all cells. Attach the finite complexes $M(f_e)$ from step 1.2 to $Z_{d-1}$ along their targets $Q_e$; call the resulting space $W$. Their retractions onto their targets glue to a strong deformation retraction $W\to Z_{d-1}$ fixing $Z_{d-1}$, because their interiors and source copies are disjoint and each homotopy fixes its target. The old retraction homotopy, the simplicial approximation homotopy and the source homotopy in step 1.2 concatenate to a homotopy in $W$ from $a_e$ to the new source-sphere inclusion $j_e$.

Continue the general metric-space argument of step 1.3, retaining its $Z$, $U_Z$, ${\mathrm{\Lambda }}_Z$ and $\mathcal{W}$. Let $N$ be the nerve of $\mathcal{W}$, with its weak simplicial topology. Choose by [F7] a locally finite subordinate partition of unity $(p_W{)}_{W\in \mathcal{W}}$ on $Z$. If an initial partition is indexed by a refinement, group the functions according to their assigned $W$; local finiteness makes each grouped sum continuous and the grouped cozero family locally finite. The positive-coordinate members at $a$ all contain $a$, so $(p_W(a){)}_W$ defines a point $p(a)\in \mathrm{\mid }N\mathrm{\mid }$. Locally only finitely many coordinates occur, proving continuity to the weak realization. Select $z_W\in W$ for each $W$ and well-order the vertices, using AC. Define $q:\mathrm{\mid }N\mathrm{\mid }\to Z$ recursively over simplices: on a vertex put $q(W)=z_W$; if the ordered simplex has first vertex $W_0$ and opposite face $F$, put $q((1-t)W_0+ty)={\mathrm{\Lambda }}_Z(z_{W_0},q(y),t)$ for $y\in F$. For a simplex choose just one point $a$ in its nonempty intersection. Every representative $z_W$ of its vertices lies in the union of the cover members containing $a$, hence in a convex $V$ from step 1.3. Induction therefore shows all expressions are defined and all face images stay in $V$. The formula agrees on faces by the vertex ordering. At the first vertex continuity follows from continuity of ${\mathrm{\Lambda }}_Z$ on the compact opposite face times the interval and its value independent of $y$ at $t=0$; thus the cone formula descends continuously. Continuity over all of $\mathrm{\mid }N\mathrm{\mid }$ follows from its weak simplex topology.

Any space $Z$ homotopy dominated by a CW complex has CW homotopy type. Indeed, write the domination as $p:Z\to N$, $q:N\to Z$ with $qp\simeq {\mathrm{id}}_Z$ and $N$ CW. Choose a CW approximation $c:C\to Z$ by [F3]. By step 2.1 lift $q$ up to homotopy to $q^{\mathrm{\prime }}:N\to C$, so $c{q}^{\mathrm{\prime }}\simeq q$. Put $r=q^{\mathrm{\prime }}p:Z\to C$. Then $cr\simeq qp\simeq {\mathrm{id}}_Z$. Further, $c(rc)\simeq c$; injectivity in step 2.1, now with CW source $C$, gives $rc\simeq {\mathrm{id}}_C$. These are both inverse identities, so $c$ is a homotopy equivalence. This argument places no separation or CW hypothesis on the dominated target $Z$.

Continue the simplicial-model construction of step 2.2. Attach one $D^d\times I$ to $W$ along each of these side homotopies $S^{d-1}\times I\to W$, retaining its bottom and top disks, and call the result $Z_d$. The radial retraction of the disk prism onto its side and bottom, respectively side and top, fixes that subset throughout and descends through this attachment. Explicitly project from $(0,-1)$ to the side and top of $D^d\times [0,1]$ and interpolate linearly; reflect the interval for the other end. The resulting retractions are continuous by the quotient product assertion [F2], also for an arbitrary disjoint family of prisms. On the bottom end, first retract the $M(f_e)$ pieces onto their targets, then use the old strong retraction $Z_{d-1}\to X^{d-1}$, extending both homotopies by the identity on the new bottom disks; their attaching boundaries lie in the fixed old $X^{d-1}$. This gives a strong retraction onto $X^d$. On the top end extend the old strong retraction $Z_{d-1}\to K_{d-1}$ by the identity on the $M(f_e)$ pieces and top disks; it fixes each attaching target $Q_e\subset K_{d-1}$. This gives a strong retraction onto the simplicial complex $K_d=K_{d-1}{\cup }_e(M(f_e)\cup C(\mathrm{sd}{P}_e))$. These endpoint retractions restrict on $Z_{d-1}$ to their old endpoints, as required. The common spaces $Z_d$ need not themselves be CW: only their explicit quotient homotopies, and the CW structures on the two retracts, are used.

Each new finite piece in $K_d$ has dimension at most $d$: in step 1.2 the piece over a source $k$-simplex is a cone on a complex of dimension at most $k$, since its image face has dimension at most $k$, and $k\le d-1$. Thus $K={\bigcup }_d{K}_d$ is a simplicial complex with $K_d$ a subcomplex, using the weak realization topology. Restrict the retraction to $X^d$ in step 3.2 to $K_d$ to obtain compatible maps $r_d:\mathrm{\mid }{K}_d\mathrm{\mid }\to X^d$. They are homotopy equivalences, as both inclusions into $Z_d$ have strong retractions. Their union $r:\mathrm{\mid }K\mathrm{\mid }\to X$ is continuous by the weak simplex topology. It is a weak equivalence: any sphere, path or disk map into either CW complex has finite cell support by [F4] and hence lies in a finite stage; any nullhomotopy also has finite support and lies in some larger stage. Surjectivity and injectivity at every source basepoint therefore follow from the homotopy equivalence $r_d$ at a common sufficiently large stage. The same argument with points and paths gives component bijectivity. By [F3], $r$ is a homotopy equivalence. This proves the simplicial-model assertion for arbitrary sets of cells and unbounded dimension.

Give the same simplicial realization $K$ the metric $d(a,b)={\sum }_v\mathrm{\mid }{a}_v-b_v\mathrm{\mid }$ on its finite barycentric coordinate vectors; write it as $K_s$, and write $K_w$ for its weak realization. The identity $i:K_w\to K_s$ is continuous simplexwise. Put $m(a)={\max }_v{a}_v$ and $w_v(a)=\max (a_v-m(a)\mathrm{/}2,0)$, and define $p(a{)}_v=w_v(a)\mathrm{/}({\sum }_u{w}_u(a))$. The denominator is positive since a maximal coordinate is retained. The functions $m,w_v$ are continuous in the metric (the maximum is 1-Lipschitz), and their nonzero sets form a locally finite family: at a point $a$ its finite support $S$ has $m(a)>0$, and in a sufficiently small metric neighborhood every coordinate outside $S$ is smaller than half the nearby maximum. Thus $p:K_s\to K_w$ is continuous: locally it lands in the finite simplex on a subset of $S$ and has continuous coordinates. Its support is contained in the support of $a$. The straight-line paths from $a$ to $p(a)$ therefore stay in that simplex. They give $ip\simeq {\mathrm{id}}_{K_s}$ by the metric formulas, and $pi\simeq {\mathrm{id}}_{K_w}$ by the weak simplex product test [F2]. Both maps and homotopies fix every vertex. Hence $K_s$ is a metric simplicial homotopy model for $X$.

On $K_s$ let $V_v=\{a:a_v>0\}$ be the open vertex stars and $U={\bigcup }_v{V}_v\times V_v$. For $(a,b)\in U$ put $c(a,b{)}_v=\min (a_v,b_v)\mathrm{/}({\sum }_u\min (a_u,b_u))$. The denominator is positive. Coordinatewise minimum is Lipschitz in the ${\mathrm{\ell }}^1$ norm, so $c$ is continuous on $U$; its support lies in the intersection of the two supports and hence is a face of both simplices. Define $\mathrm{\Lambda }(a,b,t)$ by the line segment from $a$ to $c(a,b)$ during the first half-interval and from $c(a,b)$ to $b$ during the second half. This is continuous, starts at $a$, ends at $b$, and satisfies $\mathrm{\Lambda }(a,a,t)=a$. Moreover if $a,b\in V_v$, then $c(a,b{)}_v>0$, so the entire path stays in $V_v$. We call such sets convex for this operation: $V_v\times V_v\subset U$ and $\mathrm{\Lambda }(V_v\times V_v\times I)\subset V_v$. No linear convexity across different simplices is being assumed.

Fix a vertex $b$ of $K$ and let $Z=\mathrm{\Omega }(K_s,b)$ with the ordinary compact-open topology. This topology is the uniform metric topology, for the bounded target metric of step 5.1: a uniform ball contains a finite intersection of compact-open neighborhoods, by a finite subdivision of the compact interval and continuity of the given loop; conversely each compact-open neighborhood contains a sufficiently small uniform ball by a finite subcover of its compact test set. Put $U_Z=\{(\alpha ,\beta ):(\alpha (t),\beta (t))\in U\text{ for every }t\in I\}$. It is open: the adjoint evaluation map is continuous by [F2], and a tube about the compact parameter $I$ gives the required neighborhood of any such pair. Define ${\mathrm{\Lambda }}_Z(\alpha ,\beta ,s)(t)=\mathrm{\Lambda }(\alpha (t),\beta (t),s)$. The interval exponential law [F2] proves continuity. The resulting path is based since $\mathrm{\Lambda }(b,b,s)=b$, and it fixes equal loops. Every loop $\alpha$ has a finite subdivision into closed intervals $J_1,\dots ,J_k$ with $\alpha (J_i)\subset V_{v_i}$: pull the stars back to an open cover of $I$, take a finite subcover and a sufficiently fine subdivision using the usual positive Lebesgue number for a finite interval cover (otherwise a sequence of uncovered shrinking intervals has a convergent subsequence and contradicts openness at its limit). The open neighborhood $\{\beta :\beta (J_i)\subset V_{v_i}\text{ for all }i\}$ is convex for ${\mathrm{\Lambda }}_Z$, by the corresponding star property in step 6.1. Thus the metric space $Z$ has an open cover by convex sets for a single continuous operation on a neighborhood of its diagonal.

First conclude the general metric-space argument of steps 1.3 and 2.3, with their $Z$, $U_Z$, ${\mathrm{\Lambda }}_Z$ and maps $p,q$. For each $a\in Z$, the simplex supporting $p(a)$ uses only $W$ that contain $a$. Choose the convex $V$ from step 1.3 for that point. Its representatives and $a$ lie in $V$; the recursive formula in step 2.3 therefore gives $qp(a)\in V$. Hence $(a,qp(a))$ lies in the domain of ${\mathrm{\Lambda }}_Z$, and $H(a,t)={\mathrm{\Lambda }}_Z(a,qp(a),t)$ is a continuous homotopy from ${\mathrm{id}}_Z$ to $qp$. The single continuous operation, rather than the pointwise choice of $V$, defines this homotopy; no continuous choice of cover members is needed. Thus $Z$ is homotopy dominated by the CW complex $\mathrm{\mid }N\mathrm{\mid }$. Step 3.1 makes it homotopy equivalent to a CW complex. Now apply this general conclusion to the metric loop space and its explicitly constructed operation in step 7.1. It proves CW homotopy type for $\mathrm{\Omega }(K_s,b)$.

The connected CW complex $X$ is path connected: each characteristic closed disk has path connected image, so the inverse image of any path component under each characteristic map is either the entire disk or empty. The weak topology therefore makes every path component open. Connectedness forces there to be only one. Step 4.1 gives a bijection on path components, so $K_w$ is also path connected; its surjective continuous identity map to $K_s$ makes $K_s$ path connected. Steps 4.1 and 5.1 give a homotopy equivalence between $X$ and $K_s$. Pick a vertex $b$ in the latter and a path from $b$ to the image of $x$; path connectedness supplies the path. Step 1.4 compares their based loop spaces by a homotopy equivalence. Together with step 8.1 this proves the ordinary compact-open assertion. For the compactly generated refinement, all displayed loop maps and homotopies also lift to the refinements by [F2]. A metric space is already compactly generated: if a subset is not closed, choose a sequence in it converging to a point outside it, using balls of radii $1\mathrm{/}n$ and AC; that convergent sequence with its limit is a compact Hausdorff test space and detects the failure of closedness. Thus the metric loop space in step 7.1 is unchanged by refinement. It follows that its CW-type conclusion also holds for the refined loop space of $X$. No countability or local finiteness of the original CW structure was used; AC was spent on cellwise lifts, finite approximations for arbitrary families, Whitehead, metric refinement and partition witnesses, and vertex representatives and ordering in the nerve.