Obstruction Theory, Postnikov Towers, and Classifying Spaces

Suppose $\bar{f}:B\to Y$ extends $f$. Its restriction to the characteristic disk, composed with a radial contraction of $D^{n+1}$ to its center, is a nullhomotopy of $\bar{f}{\mathrm{\mid }}_{S^n}=f\alpha$.

Conversely, let $H:S^n\times I\to Y$ satisfy $H(-,0)=f\alpha$ and have constant terminal map. Collapsing $S^n\times \{1\}$ turns the cylinder into $C{S}^n\cong D^{n+1}$, and [F2] makes $H$ descend to a map $F:D^{n+1}\to Y$ with $F{\mathrm{\mid }}_{S^n}=f\alpha$.

The maps $f$ on $A$ and $F$ on $D^{n+1}$ agree on the attaching boundary. By [F1]'s pushout universal property they glue uniquely to a continuous map $\bar{f}:B\to Y$ extending $f$. The two constructions are inverse existence implications and require no choice. $\mathrm{\square }$

First suppose $n\ge 2$. Work in one component and in a supplied universal-cover coordinate system. The lift of $X^n\cup A$ is simply connected: adjoining the remaining relative cells, whose dimensions are at least three, does not change ${\pi }_1$. Hence [F1] identifies each lifted $(n+1)$-cell characteristic class with its oriented relative cellular generator.

Now suppose $n=1$. Put $B=X^1\cup A$ and $W=X^2\cup A$. On each component $B_i$ with its supplied basepoint and whisker, $f_{\ast }:{\pi }_1(B_i)\to {\pi }_1(Y,f(b_i))$ has abelian target by hypothesis. Thus [F4] gives a unique homomorphism ${\lambda }_i:H_1(B_i;\mathbb{Z})\to {\pi }_1(Y,f(b_i))$ with $f_{\ast }={\lambda }_i\circ h_{B_i}$. For an oriented relative two-cell $e$, let $u_e\in H_2(W,B;\mathbb{Z})$ be its characteristic disk class. Its pair boundary $\mathrm{\partial }{u}_e\in H_1(B_i;\mathbb{Z})$ is the Hurewicz class of the attaching loop, including the supplied orientation and whisker. Therefore $\theta (f)(e)={\lambda }_i(\mathrm{\partial }{u}_e)$. The assumed trivial conjugation action makes this formula independent of loop transport in the target and makes the $n=1$ coefficient system constant in these component coordinates. No representatives are selected simultaneously.

In this $n\ge 2$ case, the geometric obstruction on a lifted $(n+1)$-cell is the value on its cellular generator of the composite “inverse relative Hurewicz, relative boundary, then $f_{\ast }$,” with the prescribed whisker transport. The coordinate rule is equivariant under deck transformations, so it descends to the local cochain $\theta (f)$ of the definition.

For $n=1$, let $c$ be an oriented relative three-cell. Its attaching sphere determines $v_c\in H_2(W;\mathbb{Z})$; under the relative inclusion $i_{\ast }:H_2(W;\mathbb{Z})\to H_2(W,B;\mathbb{Z})$, the class $i_{\ast }{v}_c$ is the relative cellular boundary of $c$, by the connecting-map and signed-incidence description in [F3]. The sphere and all its boundary incidences lie in one component, so Step 1.2 gives $(\delta \theta (f))(c)={\lambda }_i\left(\mathrm{\partial }{i}_{\ast }{v}_c\right)=0.$ The last equality is exactness of the pair homology sequence [F5]. This argument retains every cell and path in the arbitrary subcomplex $A$ inside the pair $(W,B)$; it never assumes that $A$ is simply connected.

For $n\ge 2$, let $c$ be an oriented lifted $(n+2)$-cell. Naturality of relative Hurewicz for the two consecutive skeletal pairs identifies the cellular boundary of $c$ with the Hurewicz image of its attaching class in ${\pi }_{n+1}(X^{n+1}\cup A,X^n\cup A)$. Evaluating $\theta (f)$ on that boundary is therefore $f_{\ast }$ applied after the next relative boundary. The consecutive maps ${\pi }_{n+1}(X^{n+1}\cup A)\to {\pi }_{n+1}(X^{n+1}\cup A,X^n\cup A)\to {\pi }_n(X^n\cup A)$ have zero composite by [F2]. Thus $(\delta \theta (f))(c)=0$.

The relative $(n+2)$-cells freely generate the cellular chain module, so Steps 3.1 and 2.2 give $\delta \theta (f)=0$ in their respective ranges, component by component. For $n\ge 2$, [F3] includes exactly the whisker monodromy used in Step 2.1; for $n=1$, it is the identity by Step 1.2. Hence $\theta (f)$ defines a cellular cohomology class, and the AT-23 comparison in [F3] carries it naturally to the stated singular local-coefficient class. No simultaneous choices beyond the supplied coordinates are made. $\mathrm{\square }$

An orientation reversal multiplies both the cellular generator and the recorded obstruction value by $-1$, while a lift change applies the same deck transformation and inverse monodromy relation on the equivariant cellular Hom; these are exactly the canonical basis identifications in [F1]. If a whisker $u$ from the attaching-sphere basepoint $a$ to the chosen coordinate $c$ is replaced by $v:a\to c$, the comparison loop at $c$ is $\bar{u}\ast v$, not $u\ast \bar{v}$ (which is based at $a$). Writing $fu$ and $fv$ for their images in $Y$, the published convention gives $T_u={\beta }_{\overline{fu}}$ and ${\beta }_{\gamma \ast \eta }={\beta }_{\gamma }{\beta }_{\eta }$, so $T_u={\beta }_{\overline{fu}\ast (fv)}{T}_v$. Thus this coordinate-loop transport carries the new recorded value to the old one; its inverse ${\beta }_{\bar{v}\ast u}$ carries old to new. In the $n=1$ case the action is trivial by the standing coefficient hypothesis. Finally, changing transport coordinates means applying a stalkwise natural isomorphism of the fixed local system in [F5]. By the defining naturality square it intertwines transport along every incidence path. Applying it stalkwise therefore commutes with the cellular coboundary and sends each old obstruction value to its new coordinate. In all four cases the resulting canonical cellular-cochain isomorphism carries $\theta (f)$ and its cohomology class to their new-coordinate versions.

Give $(X\times I,A\times I)$ the relative prism CW structure and put

Evaluate $\delta \mathrm{\Theta }=0$ on $e^{n+1}\times I$. Using [F3] and the defining sign $d(e^n)=(-1{)}^{n+1}\mathrm{\Theta }(e^n\times I)$ gives $0=(-1{)}^{n+1}(\delta d(e^{n+1})+\theta (f_1)(e^{n+1})-\theta (f_0)(e^{n+1}))$. Hence $\delta d=\theta (f_0)-\theta (f_1)$ on every cell.

Coboundaries vanish in cohomology, so Step 2.1 identifies $[\theta (f_0)]$ and $[\theta (f_1)]$ after the coefficient transport supplied by $H$. Combining this with Step 1.1 proves independence from every listed choice. The argument uses a supplied homotopy and supplied coordinates and makes no set-indexed selection. $\mathrm{\square }$

Suppose the prior-stage restriction extends to $F:X^{n+1}\cup A\to Y$. Put $f_1=F{\mathrm{\mid }}_{X^n\cup A}$. Every attaching sphere then bounds its characteristic-disk restriction, so $\theta (f_1)=0$ by [F3]. The independence theorem, applied to the relevant prior-stage homotopy, gives $[\theta (f)]=[\theta (f_1)]=0$.

Let $d\in C^n(X,A;\mathcal{P})$ and consider one oriented relative $n$-cell with characteristic disk $D^n$ and cell map $f_e$. Choose a based sphere map $u_e:S^n\to Y$ representing the sign-adjusted value $d(e)$ in the stalk fixed by the cell's whisker. There is a relative pinch map $q:D^n\to D^n\vee S^n$: choose a small closed ball in the interior, collapse its boundary to the wedge point, map the outside quotient to the first disk by a radial homeomorphism fixed on $\mathrm{\partial }{D}^n$, and map the collapsed inner ball with degree $+1$ to the sphere summand. Define $f_{d,e}=(f_e\vee u_e)q$. It agrees with $f_e$ on $\mathrm{\partial }{D}^n$. In the prism-boundary sphere of [F2], collapse the stationary side and the region on which the two disk maps agree. What remains is exactly the degree-one sphere carrying $u_e$; choosing the sign of $u_e$ according to [F2]'s convention therefore gives $d(f,\mathrm{const},f_d)(e)=d(e)$.

Apply Step 1.2 to every relative $n$-cell. AC in [A1] selects the sphere representatives for an arbitrary family; only finitely many choices occur for a finite pair. The modified cell maps agree with the unchanged map on $X^{n-1}\cup A$, so the CW pushout and weak topology glue them to $f_d:X^n\cup A\to Y$, with $d(f,\mathrm{const},f_d)=d$. No homotopy from $f$ to $f_d$ on the $n$-cells is constructed or needed.

Conversely, assume $[\theta (f)]=0$ and choose $d$ with $\theta (f)=\delta d$. Apply Step 2.1 and put $f^{\mathrm{\prime }}=f_d$. By [F1], $\theta (f)-\theta (f^{\mathrm{\prime }})=\delta d=\theta (f)$, hence $\theta (f^{\mathrm{\prime }})=0$. By [F3], $f^{\mathrm{\prime }}$ extends over every relative $(n+1)$-cell. AC selects all fillers for an arbitrary cell family, and the pushout glues them to an extension on $X^{n+1}\cup A$. This extension restricts to the original map on $X^{n-1}\cup A$.

Steps 1.2–2.1 also prove the more precise realization assertion, and Steps 1.1 and 3.1 prove both implications. Zero cochains, absent cells, and the case $A=X$ are included. The proof does not say that the original fixed map $f$ extends when merely its cohomology class vanishes: it may first be changed, generally nonhomotopically rel boundary, on the $n$-cells while staying fixed on the prior skeleton. $\mathrm{\square }$

Let $g_0,g_1\in {\mathcal{E}}_n(a)$. By [F1], choose a homotopy $H$ rel $A$ between their restrictions to $X^{n-1}\cup A$. Because $\theta (g_0)=\theta (g_1)=0$, [F2] gives $\delta d(g_0,H,g_1)=0$. Thus the difference cochain defines a class in $H^n(X,A;\mathrm{\Pi })$.

Changing $H$ or changing either endpoint through a homotopy rel $A$ changes this cocycle by a coboundary: apply the obstruction-class independence theorem to the corresponding boundary map on the product pair. Hence $[d(g_0,g_1)]$ depends only on the two classes in ${\mathcal{E}}_n(a)$. Reversing a prism changes its sign, and gluing prisms gives

Regard the homotopy problem as extension over the product pair in [F3]. Suppose $[d(g_0,H,g_1)]=0$, and choose $c\in C^{n-1}(X,A;\mathrm{\Pi })$ with $d(g_0,H,g_1)=\delta c$. Keep both endpoint maps fixed. On each relative prism $e^{n-1}\times I$, whose dimension is $n$, insert by the relative pinch construction of [F4] a sphere representative of $-c(e^{n-1})$ into the interior of $H$, leaving its entire boundary, including the two endpoint faces, fixed. AC makes these simultaneous insertions over arbitrary cells; the CW pushout glues them to a new prior-stage homotopy $H_c$ rel $A$ with the same endpoints. On a boundary prism $e^n\times I$, the only changed faces are the prisms over the $(n-1)$-cells of $\mathrm{\partial }{e}^n$. Their signed incidence sum, with the interval-last sign in [F3], is $-\delta c(e^n)$; the same oriented boundary calculation as [F2] therefore gives $d(g_0,H_c,g_1)=d(g_0,H,g_1)-\delta c=0$ as a cochain, not just as a class. The zero obstruction on each $e^n\times I$ now supplies a filling extending $H_c$ over the relative $n$-prisms, while the fixed bottom and top faces remain $g_0$ and $g_1$. The glued fillings are a homotopy $g_0\simeq g_1$ on $X^n\cup A$ rel $A$. Conversely, such a homotopy fills every prism, making its difference cochain zero and hence its class zero.

Fix $g\in {\mathcal{E}}_n(a)$ and let $z\in Z^n(X,A;\mathrm{\Pi })$. Since $g$ has zero obstruction as in Step 1.1, apply [F4] to a stationary prior-stage homotopy and prescribe difference cochain $z$. It produces $g_z:X^n\cup A\to Y$ with $d(g,\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},g_z)=z$. By [F2],

If $z$ and $z^{\mathrm{\prime }}$ differ by a coboundary, Step 1.2 gives $[d(g_z,g_{z^{\mathrm{\prime }}})]=[z^{\mathrm{\prime }}-z]=0$, so Step 2.1 makes $g_z$ and $g_{z^{\mathrm{\prime }}}$ equivalent. Thus Step 2.2 defines an action of $H^n(X,A;\mathrm{\Pi })$ on ${\mathcal{E}}_n(a)$. The addition formula in Step 1.2 proves the action law.

For any $g_0,g_1$, the class $[d(g_0,g_1)]$ sends $g_0$ to $g_1$, proving transitivity. If a class fixes $g_0$, its displacement is zero by Step 2.1, proving freeness. AC enters only in [F4] and in simultaneous extension over arbitrary cell families; finite families need only finite choice. Empty cell sets, $A=X$, and the zero group give the asserted singleton torsors. $\mathrm{\square }$

Form $f^{\ast }E=\{(x,e):f(x)=p(e)\}$ with projection $q(x,e)=x$. The map $g$ corresponds to the section $s(x)=(x,g(x))$, and conversely a section has second coordinate a lift. Pullbacks preserve the Serre lifting property.

Construct the coefficient system for the varying fibers. For a base path $\gamma :b\to c$ and a chosen lift $\tilde{\gamma }$ beginning at $e\in F_b$ and ending at $e^{\mathrm{\prime }}\in F_c$, lift the constant-in-the-sphere-coordinate homotopy $\gamma$ on $S^n\times I$, prescribed on $S^n\times \{0\}\cup \{\ast \}\times I$ by a based sphere map $a:S^n\to F_b$ and $\tilde{\gamma }$. The top face gives a based sphere map into $(F_c,e^{\mathrm{\prime }})$. Relative lifting on a second parameter cube shows that homotopic based sphere representatives give homotopic top faces, while lifting the two halves of the cubical concatenation and comparing along their common face shows preservation of the ${\pi }_n$ group law. Reversing $\gamma$ gives an inverse up to the based retracing-prism homotopy, so this is an isomorphism ${\pi }_n(F_b,e)\to {\pi }_n(F_c,e^{\mathrm{\prime }})$. The same relative lifting on a square compares two path lifts and an endpoint-fixed homotopy of base paths; the top edge of that square is a path between their endpoint basepoints in $F_c$. Correcting by its basepoint transport from [F2] makes the maps agree. A two-interval prism compares a concatenated path with successive transports. Because the fiber is simple in degree $n$, loops in a fiber act trivially on ${\pi }_n$, so the correction does not depend on the comparison edge; for $n=1$ this is precisely the stated abelian/trivial-conjugation condition. The resulting maps depend only on endpoint-fixed base-path classes, preserve composition, and are invertible. Consequently $b\mapsto {\pi }_n(F_b)$ is an abelian local system ${\mathrm{\Pi }}_{n}F$ on the relevant base component, and pulling it back along $f$ gives the stated $f^{\ast }{\mathrm{\Pi }}_{n}F$. This uses only finite cubical lifting for each supplied path; [A1] is needed later for simultaneous choices over arbitrary cells.

Let $\varphi :(D^{n+1},S^n)\to (X^{n+1}\cup A,X^n\cup A)$ be a characteristic map for the pullback section of Step 1.1. Contract $D^{n+1}$ to its center and lift that contraction along $q$ on the boundary section. The terminal boundary map lies in the fiber over the center and defines

Choose orientations, lifts of relative cells, and whiskers. Step 2.1 assigns a value to every relative $(n+1)$-cell. Replacing a whisker by a loop applies precisely the fiber-transport automorphism of [F2], while a deck translate applies the corresponding equivariance rule. The values therefore form a well-typed cellular cochain

Evaluate the cochain of Step 3.1 on the boundary of one relative $(n+2)$-cell. Pull everything back to its characteristic disk. Lifting its radial contraction identifies all boundary fiber groups, and the signed incidence sum is the boundary of the single lifted sphere datum on that disk. It is zero in ${\pi }_n(F)$ because a boundary is null in the relative homotopy exact sequence. Undoing the transports restores exactly the local-coefficient incidence formula. Hence $\delta {\theta }_{n+1}(s)=0$.

A different cellular contraction, whisker, or partial section gives a fiberwise prism. Applying the signed boundary calculation of Step 4.1 to that prism yields the identity recorded in [F5]:

If the class from Step 5.1 is zero, write ${\theta }_{n+1}(s)=\delta d$. On a relative $n$-cell, lift a contraction of its base disk to transport the given section to a map into the center fiber. Apply the relative pinch construction of [F6] there, inserting a signed sphere representative of $d(e^n)$ while fixing the boundary. Lift the reversed contraction relative to the boundary; the resulting map is again a section over the cell and agrees with $s$ on $X^{n-1}\cup A$. AC chooses the sphere representatives and relative lifts simultaneously over all cells. The fiberwise prism calculation of Step 5.1 then gives ${\theta }_{n+1}(s_d)={\theta }_{n+1}(s)-\delta d=0$. Step 2.1 extends $s_d$ over every relative $(n+1)$-cell, and the CW pushout glues the extensions. Conversely, an extended section has zero cell values, so its class, and hence the class of the original partial lift, is zero.

If ${\pi }_i(F)=0$ for $i<n$, every earlier cell obstruction group is zero. Induction gives a lift through the $n$-skeleton, and the same prism identity shows that any two such lifts have the same primary class. If the original map is a numerable bundle projection, the published theorem makes it a Hurewicz and therefore a Serre fibration, so all preceding steps apply. AC is confined to [A1], and finite cell families require only finite choice. $\mathrm{\square }$

For $G$, begin with a wedge $W$ of one oriented circle $x_g$ for every $g\in G$. Attach a $2$-cell along the word $x_g{x}_h{x}_{gh}^{-1}$ for every ordered pair $(g,h)$. By [F1], the resulting complex $Q_2$ has presentation

Now let $n\ge 2$. Put $W={\bigvee }_{a\in A}{S}_a^n$. By [F2], ${\pi }_n(W)={\mathbb{Z}}^{(A)}$. Let $ϵ:{\mathbb{Z}}^{(A)}\to A$ send the basis vector $e_a$ to $a$. Choose a sphere representative for every element of $\ker ϵ$ and attach an $(n+1)$-cell along it, obtaining $Q_{n+1}$. The pair $(Q_{n+1},W)$ is $n$-connected and $W$ is simply connected. Relative Hurewicz identifies its relative ${\pi }_{n+1}$ with the free relative cell group, and the boundary map sends each cell generator to its attaching class. Exactness therefore gives

Starting from the degree-one presentation in Step 1.1, inductively choose one based map $S^r\to Q_r$ representing every element of ${\pi }_r(Q_r)$ and attach an $(r+1)$-cell along each, for $r\ge 2$. The relative exact sequence makes ${\pi }_r(Q_r)\to {\pi }_r(Q_{r+1})$ zero and surjective, hence ${\pi }_r(Q_{r+1})=0$, while [F3] preserves all lower groups. Put $Q={\bigcup }_{r\ge 2}{Q}_r$. For fixed $i>1$, later cells do not recreate ${\pi }_i$. By [F4], every representative and nullhomotopy in $Q$ occurs at a finite stage; consequently ${\pi }_1(Q)=G$ and ${\pi }_i(Q)=0$ for $i>1$. Thus $Q$ is a $K(G,1)$.

Starting from the degree-$n$ complex in Step 1.2, attach one $(r+1)$-cell along a representative of every element of ${\pi }_r$ at the current stage, beginning with $r=n+1$. The argument of Step 2.1, now preserving ${\pi }_n=A$, kills each higher group successively. The increasing union $Q$ has ${\pi }_n(Q)=A$ and every other positive homotopy group zero by [F4]. This is a $K(A,n)$.

Let $K$ be any other degree-one model with the same identified group. From the completed model in Step 2.1, map each circle $x_g$ to a based loop representing the corresponding $g\in {\pi }_1(K)$. Each multiplication relator maps to a nullhomotopic loop, so choose fillings of the $2$-cells. Every higher attaching sphere maps trivially because ${\pi }_r(K)=0$ for $r>1$, and induction extends the map to $u:Q\to K$. It induces the prescribed isomorphism on ${\pi }_1$.

For a degree-$n$ model $K$, begin with the completed model in Step 3.1 and map the sphere indexed by $a$ to a representative of the corresponding element of ${\pi }_n(K)$. Every attaching map indexed by $\ker ϵ$ becomes nullhomotopic, so the map extends over the $(n+1)$-cells. All later attaching maps extend because the corresponding higher homotopy groups of $K$ vanish. The resulting $u:Q\to K$ induces the prescribed isomorphism on ${\pi }_n$.

In either case, $Q$ and $K$ are connected, $u$ is an isomorphism on their sole possibly nonzero positive homotopy group, and all their other positive homotopy groups vanish. Thus $u$ is a weak equivalence. By [F5], it is a homotopy equivalence. Applying Steps 3.2 and 4.1 to two models gives a zigzag of homotopy equivalences through $Q$; choosing a homotopy inverse for one leg gives a homotopy equivalence between the models. Its induced group map is the prescribed identification, with the basepoint track supplying the standard conjugacy transport when $n=1$.

The construction also covers the trivial group. In that case the resulting connected CW complex has every positive homotopy group zero, and its map to a point is a weak equivalence, hence a homotopy equivalence by [F5]. No countability, finite generation, or finite-dimensionality has been assumed. Every infinite selection is accounted for by [A1], while the passage to the union uses the individual compact-support statement [F4], not an unproved interchange of homotopy groups with an arbitrary colimit. $\mathrm{\square }$

By [F1], identify $H_n(K)$ with $A$. In the UCT exact sequence, the group to the left of evaluation vanishes for the reasons in [F2]. Therefore evaluation is an isomorphism, and there is a unique class $\iota$ satisfying