Blakers--Massey connectivity for a homotopy-pushout square

For a map $j:X\to Y$ and $s\ge 1$, the mapping-cylinder pair sequence [F4] shows that $s$-connectivity is equivalent to component bijectivity, isomorphisms on ${\pi }_i$ for $1\le i<s$, and surjectivity on ${\pi }_s$, all at source basepoints. In the forward direction the two adjacent relative terms give each isomorphism, and the following relative term gives surjectivity; trivial relative degree one makes two source components joined in the cylinder already equal, by its pointed tail. Conversely, for relative degree $i\ge 2$ with $i\le s$, a relative class has boundary in the zero kernel of the preceding absolute map and hence comes from the ambient absolute group; surjectivity in that degree makes this image zero. In degree one, component injectivity and the pointed tail put each relative class in the image of the ambient fundamental group, and its surjectivity from the source makes that class distinguished. The cylinder deformation gives the absolute groups of $Y$, with its actual basepoint track by [F8]. Consequently this criterion is preserved by homotopies of maps and by homotopy equivalences on their source or target, using [F8] at the relevant track and ordinary paths on components.

Every CW complex is CGWH in the conventions of [F2]. Indeed a subset whose inverse image under every compact Hausdorff test map is closed has closed inverse image under each characteristic disk, by disk compactness in [F9]; the CW weak topology then makes it closed. Conversely closed subsets pass all continuous tests. Hausdorffness and closedness of compact images give the WH condition. A CW path component is an open and closed subcomplex: each cell closure is path connected and belongs to one component; each component and its complement therefore have inverse image either the whole disk or the empty set under every characteristic map, making both closed. Thus we may restrict a CW complex to one component without losing its CW structure or topology.

First prove the conclusion for a CW union $D=U\cup V$ with $A=U\cap V$ nonempty path connected, $(U,A)$ $m$-connected and $(V,A)$ $n$-connected. These component conditions make $U,V,D$ path connected. Put $H=U{\times }_D^{h}V$ and fix $a\in A$. Let $E=\{(x,\gamma ):x\in A,\gamma :I\to V,\gamma (0)=x\}$ with projection $p(x,\gamma )=\gamma (1)$. The projection $q:H\to V$ sends $(u,\omega ,v)$ to $v$. It is the pullback of the mapping-path fibration for $U\hookrightarrow D$: its lift for any prescribed homotopy in $V$ is obtained by composing that homotopy with $V\hookrightarrow D$ in the explicit lifting formula of [F3], retaining its given $V$ coordinate. Thus both $p,q$ are Hurewicz, hence Serre, fibrations. The continuous map $T:E\to H,T(x,\gamma )=(x,\gamma ,\gamma (1))$ lies over the identity of $V$. The constant-path inclusion $k:A\to E$ is a homotopy equivalence by [F3], and $Tk(x)=(x,c_x,x)$ is the constant-path comparison for this strict CW union.

Now suppose $f,g$ are cellular. In their double cylinder $P$, let $U$ consist of $B$ and the half-cylinder $A\times [0,1\mathrm{/}2]$, and $V$ consist of $C$ and $A\times [1\mathrm{/}2,1]$. By [F7] these are ordinary CW cylinders, with common free-end subcomplex $A_0=A\times \{1\mathrm{/}2\}$. Gluing the relative cells of $V\setminus A_0$ onto $U$ in their original dimensional order satisfies the finite-support attachment hypotheses of [F7]. Its map-out test is exactly agreement of the maps on $U,V$, hence the ordinary double-cylinder quotient test; thus it gives the actual CW space $P$. The retractions $U\to B$ and $V\to C$ identify the two source inclusions with $f,g$ up to the explicit cylinder tracks. By step 1.1 the pairs $(U,A_0)$ and $(V,A_0)$ have connectivities $m,n$.

Over $a$ the fiber map is the inclusion of path models $\varphi :F_A\to F_U,F_A=\{(x,\gamma ):x\in A,\gamma (0)=x,\gamma (1)=a,\gamma (I)\subseteq V\},$ where $F_U$ has $x\in U$ and its path in $D$. Base both at $(a,c_a)$. For $j\ge 1$, a based $j$-cube in $F_A$ is precisely a map $w:I^j\times I\to V$ whose bottom face lies in $A$, top face is $a$, and side faces are $a$: the fiber coordinates are $(w(z,0),t\mapsto w(z,t))$. Transposition [F2] identifies continuous maps and homotopies in both directions. These are exactly the relative $(j+1)$-cube and homotopy equations in [F5], and cutting the first coordinate gives the same group operation. Hence ${\pi }_j(F_A,(a,c_a))\cong {\pi }_{j+1}(V,A,a),{\pi }_j(F_U,(a,c_a))\cong {\pi }_{j+1}(D,U,a)\mathrm{.}$ For $j=0$, a fiber point is a relative path, and a path in the fiber is exactly a relative path homotopy with endpoint $a$ fixed. Thus the same correspondence identifies component sets with the relative degree-one pointed sets. Under every one of these identifications, ${\varphi }_{\ast }$ is precisely the excision inclusion, since its formula keeps the same cube and only enlarges its target pair.

By [F1] and step 3.1, ${\varphi }_{\ast }$ is an isomorphism on positive ${\pi }_j$ for $j<r$, and surjective for $j=r$; on components it is bijective because $1<N$. The source component set is a singleton since $(V,A)$ is $n$-connected and $n\ge 1$, so both fibers are path connected. The total spaces $E,H$ are path connected as well: the base $V$ is path connected, every point of a total space can be joined to the fiber over $a$ by lifting a path to $a$, and that fiber is path connected and nonempty. These path lifts exist by [F3]. No path for a family of points is selected.

Compare the two fibration sequences [F4] for $T$, with identical base $V$. They commute: inclusion and projection commute pointwise, and the boundary comparison commutes because composing a lift with $T$ is a lift of the same base cube, so the lift-independent connecting class in [F4] has the same image. For $1\le i\le r$ and $y\in {\pi }_i(H)$, let $b=q_{\ast }y$. When $i\ge 2$, its boundary in ${\pi }_{i-1}(F_A)$ maps to zero under the injective ${\varphi }_{i-1}$, so is zero. When $i=1$, both fiber component sets are singletons, giving the same conclusion. Exactness supplies $x\in {\pi }_i(E)$ with $p_{\ast }x=b$. Then $y(T_{\ast }x{)}^{-1}$ lies in the image from ${\pi }_i(F_U)$. Lift its fiber preimage through the surjective ${\varphi }_i$ and multiply its image in ${\pi }_i(E)$ on the left of $x$; the result maps to $y$. Thus $T_{\ast }$ is surjective through degree $r$. For $1\le i<r$, if $T_{\ast }x=1$, its base image is one, so $x$ comes from $z\in {\pi }_i(F_A)$. The image ${\varphi }_{i}z$ comes from the boundary of some $b\in {\pi }_{i+1}(V)$, by exactness in the lower sequence. Naturality and injectivity of ${\varphi }_i$ imply $z=\mathrm{\partial }b$, whose image in ${\pi }_i(E)$ is one. Hence $x=1$. This proves injectivity below $r$, including the nonabelian degree-one case. Together with step 4.1 and the criterion of step 1.1, $T$, and then $Tk$, are $r$-connected. The point $a$ was arbitrary.

By step 1.1 the maps $f,g$ are bijective on components. Since CW components are clopen subcomplexes by step 1.2, the double cylinder splits into the clopen unions of the component of $A$, its corresponding component of $B$, and its corresponding component of $C$. Each path in $P$ stays in one such component. Hence the homotopy pullbacks split into those same clopen pieces: a triple with its endpoints in different pieces has no intervening path. On each piece the intersection $A_0$ is nonempty path connected, so steps 2.1–5.1 apply to show that ${\eta }_0:A_0\to U{\times }_P^{h}V,x\mapsto (x,c_x,x)$ is $r$-connected. There is one target component for each component of $A$ by step 4.1. Thus it is $r$-connected on the whole space, with the required component bijection and groups at every source point. This argument treats one component at a time and chooses no representatives of all components.

Let $D_U(u,s)$ and $D_V(v,s)$ be the cylinder deformations from the identity to the retractions $r_U:U\to B$ and $r_V:V\to C$, fixing $B,C$. Define $L:U{\times }_P^{h}V\to B{\times }_P^{h}C$ by sending $(u,\omega ,v)$ to $(r_{U}u,\overline{D_U(u,-)}\ast \omega \ast D_V(v,-),r_{V}v)$, using three equal path intervals. Its inverse up to homotopy is the endpoint-inclusion map $J$. All maps are continuous by the adjoint and quotient tests [F2]. For $JL$, a homotopy starting from the identity with harmless constant path pauses moves the endpoints to $D_U(u,s),D_V(v,s)$ and uses the path $\overline{D_U(u,-){\mathrm{\mid }}_{[0,s]}}\ast \omega \ast D_V(v,-){\mathrm{\mid }}_{[0,s]}$, each restricted track linearly parametrized on its own interval. At $s=0$ this is $c_u\ast \omega \ast c_v$, homotopic to $\omega$ by linear interpolation of the continuous nondecreasing parameter functions with fixed endpoints; at $s=1$ it is $JL$. On the smaller pullback, $LJ$ only inserts constant paths since the deformations fix $B,C$, and the same reparametrization gives the identity. Thus $L$ is a homotopy equivalence. Applied to ${\eta }_0(a,1\mathrm{/}2)$ its two half-cylinder tracks concatenate to the full cylinder path from $f(a)$ to $g(a)$, with a constant middle segment. Removing that segment gives the specified $\eta (a)$ by a homotopy with its actual moving pullback basepoint. Steps 1.1 and 6.1 therefore prove the cellular assertion for the original endpoint pullback.

For arbitrary $f,g$, use [F6] to choose cellular $f^{\mathrm{\prime }},g^{\mathrm{\prime }}$ and homotopies $\alpha :f\simeq f^{\mathrm{\prime }}$, $\beta :g\simeq g^{\mathrm{\prime }}$. Use its finite-source clause if $A$ is finite; otherwise use [A1]. These homotopies preserve the connectivities by step 1.1. Denote their double cylinder by $P^{\mathrm{\prime }}$, with cylinder path ${\sigma }_a^{\mathrm{\prime }}$. Define $\mathrm{\Phi }:P\to P^{\mathrm{\prime }}$ to be the identity on $B,C$ and to send the old cylinder path to $\alpha (a,-)\ast {\sigma }_a^{\mathrm{\prime }}\ast \overline{\beta (a,-)}\mathrm{.}$ The endpoints are exactly $f(a)$ and $g(a)$, so [F2] gives a continuous descended map. Reversing the two homotopies defines $\mathrm{\Psi }:P^{\mathrm{\prime }}\to P$, also the identity on $B,C$. Their composites on a cylinder path insert a track immediately followed by its reverse at each end. For a path $\gamma$ the backtrack $\gamma \ast \bar{\gamma }$ contracts rel endpoints by $t\mapsto \gamma ((1-s)\tau (t))$, where $\tau (t)=2t$ on the first half and $2-2t$ on the second. This formula is continuous at $s=1$ and fixes the starting endpoint. Apply it to each inserted backtrack, then remove constant pauses by parameter interpolation as in step 7.1. The resulting homotopies paste with the fixed maps on $B,C$ and descend through the cylinder quotient by [F2]. Hence $\mathrm{\Phi },\mathrm{\Psi }$ are homotopy inverses rel $B⨿C$.

Postcomposing path coordinates with $\mathrm{\Phi }$ gives a continuous map ${\mathrm{\Phi }}_H:B{\times }_P^{h}C\to B{\times }_{P^{\mathrm{\prime }}}^{h}C$. Postcomposition with $\mathrm{\Psi }$ is its homotopy inverse: the homotopies of step 8.1 fix both endpoint subspaces, so postcomposing a path with them remains in the required pullback at every time. It remains to compare the actual source maps. The triple ${\mathrm{\Phi }}_H\eta (a)$ has path ${\alpha }_a\ast {\sigma }_a^{\mathrm{\prime }}\ast {\bar{\beta }}_a$ and endpoints $f(a),g(a)$. Move these endpoints to $\alpha (a,s),\beta (a,s)$ and replace the first and last tracks by their remaining segments from parameter $s$ to one. Explicitly, on the three equal path intervals the path has values $\alpha (a,s+(1-s)3t),{\sigma }_a^{\mathrm{\prime }}(3t-1),\beta (a,s+(1-s)(3-3t)),$ respectively. The seams agree at $f^{\mathrm{\prime }}(a),g^{\mathrm{\prime }}(a)$, and the endpoints are the moving endpoints just specified. At $s=1$ it is the new cylinder path with constant end pauses; remove those pauses as in step 7.1. Thus ${\mathrm{\Phi }}_H\eta \simeq {\eta }^{\mathrm{\prime }}$, where ${\eta }^{\mathrm{\prime }}$ is the canonical comparison for $f^{\mathrm{\prime }},g^{\mathrm{\prime }}$. This verifies the source map as well as the target homotopy type. The cellular conclusion and step 1.1 now imply that $\eta$ is $r=m+n-1$ connected.

If $A$ is empty, component-surjectivity of each initial map forces $B=C=\mathrm{\varnothing }$, so both pushout and pullback are empty and the conclusion is vacuous with a bijection on empty component sets. Point spaces, an identity leg and zero relative cells are retained by the cylinder constructions. For $m=n=1$, $N=2$ and $r=1$: step 4.1 gives connected fibers and a surjection on their fundamental groups, and step 5.1 proves exactly component bijectivity and surjectivity on total fundamental groups, with no unjustified endpoint injectivity. Below the endpoint it proves both kernel and image assertions. The group shift is explicit in step 3.1: a $j$-cube of a homotopy fiber is a relative $(j+1)$-cube, giving the claimed $m+n-1$ convention. All based claims use the actual comparison image and the tracks in steps 7.1 and 9.1. The cellular proof uses no choice; the general proof invokes [A1] only in step 8.1, and finite $A$ uses the finite approximation clause there. This completes the theorem.