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Difference cochains classify homotopies of extensions in the stable stage

Statement

Assume AC. Let $(X, A)$ be a relative CW complex, let $n \geq 1$ , and let $Y$ be $(n - 1)$ -connected and $n$ -simple; when $n = 1$ , assume in particular that $π_{1} (Y)$ is abelian. Fix $a : A \to Y$ and put $Π = π_{n} (Y)$ with its resulting simple coefficient system.

Let $E_{n} (a)$ consist of maps $g : X^{n} \cup A \to Y$ extending $a$ whose obstruction cochain is zero, modulo homotopy rel $A$ on $X^{n} \cup A$ . Equivalently, these are the $n$ -stage maps which extend over $X^{n + 1} \cup A$ , with an extension chosen only when needed. If $E_{n} (a)$ is nonempty, then

$H^{n} (X, A; Π)$

acts freely and transitively on it. For $g_{0}, g_{1} \in E_{n} (a)$ , the displacement is the difference class

$[d (g_{0}, g_{1})] \in H^{n} (X, A; Π),$

and this class is zero if and only if $g_{0}$ and $g_{1}$ are homotopic rel $A$ through the $n$ -skeleton. For a finite relative CW pair, only finite choice is used.

Facts & Assumptions

[F1]

Since $Y$ is $(n - 1)$ -connected, every two extensions of $a$ are homotopic rel $A$ through $X^{n - 1} \cup A$ ; for $n = 1$ , abelianness makes the conjugation action simple.

[F2]

For a chosen prior-stage homotopy, $δ d (g_{0}, g_{1}) = θ (g_{0}) - θ (g_{1})$ (The primary obstruction class is independent of cellular choices).

[F3]

The difference-cochain construction uses one shifted prism cell for each relative cell and identifies its primary obstruction cochain with the signed difference cochain (Difference cochain between two cellular extensions).

[F4]

For every cellular $n$ -cochain $d$ , the vanishing-obstruction theorem constructs a map $f_{d}$ equal to $f$ on the prior skeleton and having prescribed difference $d (f, const, f_{d}) = d$ , by relative pinch maps and simultaneous choice; it makes no claim that $f$ and $f_{d}$ are homotopic on the $n$ -skeleton (Vanishing of the primary obstruction is equivalent to extension over the next skeleton).

[A1]

AC is used only to choose representatives and fillers for arbitrary cell families (The Axiom of Choice).

Proof

Given: $(X, A)$ , $Y$ , $a$ , and [A1] as in the statement.

1.1F1F2

Let $g_{0}, g_{1} \in E_{n} (a)$ . By [F1], choose a homotopy $H$ rel $A$ between their restrictions to $X^{n - 1} \cup A$ . Because $θ (g_{0}) = θ (g_{1}) = 0$ , [F2] gives $δ d (g_{0}, H, g_{1}) = 0$ . Thus the difference cochain defines a class in $H^{n} (X, A; Π)$ .

1.2F3

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 $E_{n} (a)$ . Reversing a prism changes its sign, and gluing prisms gives

$[d (g_{0}, g_{2})] = [d (g_{0}, g_{1})] + [d (g_{1}, g_{2})] .$

[F3]

2.1A1F2F3F4step 1.1

Regard the homotopy problem as extension over the product pair in [F3]. Suppose $[d (g_{0}, H, g_{1})] = 0$ , and choose $c \in C^{n - 1} (X, A; Π)$ with $d (g_{0}, H, g_{1}) = δ c$ . 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 $\partial e^{n}$ . Their signed incidence sum, with the interval-last sign in [F3], is $- δ c (e^{n})$ ; the same oriented boundary calculation as [F2] therefore gives $d (g_{0}, H_{c}, g_{1}) = d (g_{0}, H, g_{1}) - δ c = 0$ 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 $g_{0} ≃ g_{1}$ on $X^{n} \cup A$ rel $A$ . Conversely, such a homotopy fills every prism, making its difference cochain zero and hence its class zero.

2.2F2F4step 1.1

Fix $g \in E_{n} (a)$ and let $z \in Z^{n} (X, A; Π)$ . 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 $d (g, c o n s t, g_{z}) = z$ . By [F2],

$θ (g_{z}) = θ (g) - δ z = 0,$

so $g_{z} \in E_{n} (a)$ . [F2, F4]

3.1step 1.2step 2.1step 2.2

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

4.1A1step 2.1step 3.1

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. $□$

Difference cochains classify homotopies of extensions in the stable stage

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources