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Weak contractibility of the irreducible connection locus

Statement

Assume ZF together with the Axiom of Choice. Let $M$ be $S^{3}$ or $S^{4}$ , let $N \geq 2$ , and let $P \to M$ be any smooth principal right $S U (N)$ -bundle. Fix a smooth reference connection $A_{0}$ , a basepoint $m \in M$ , and a frame of $P_{m}$ . Equip the affine space $A_{\infty} = A_{0} + Ω^{1} (M; ad P)$ with its $C^{\infty}$ section topology, and let $A_{\infty}^{*}$ be the connections whose based holonomy action on $C^{N}$ has no nonzero proper invariant complex subspace. Then $A_{\infty}^{*}$ is nonempty and path-connected, and for every $A \in A_{\infty}^{*}$ and every integer $j \geq 1$ , $π_{j} (A_{\infty}^{*}, A) = 0$ . Here $π_{j} (X, x)$ is the based homotopy group of continuous maps $u : (I^{j}, \partial I^{j}) \to (X, x)$ modulo homotopies that keep $\partial I^{j}$ at $x$ .

Facts & Assumptions

Given: ZF and the Axiom of Choice; $M = S^{3}$ or $S^{4}$ ; $N \geq 2$ ; a smooth principal right $S U (N)$ -bundle $P$ ; a smooth reference connection $A_{0}$ ; and a based irreducible connection $A \in A_{\infty}^{*}$ when a basepoint is specified.

[A1]

The Axiom of Choice means every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

The batch-56 configuration definition sets $A_{\infty} = A_{0} + Ω^{1} (M; ad P)$ and defines $A_{\infty}^{*}$ by irreducibility of the based holonomy action on $C^{N}$ . Its complete definition was inspected and its current Step 1 record is ready; the argument uses exactly this interface (Smooth and Sobolev configuration and reduced gauge spaces).

[F2]

For every continuous map from a finite polyhedron into $A_{\infty}$ , with irreducible values on a subpolyhedron, the relative avoidance lemma supplies a homotopy fixed on that subpolyhedron and an open neighborhood of it whose endpoint is entirely irreducible. Its complete argument and exact batch-56 inputs were inspected; the configuration and stabilizer suppliers now have current ready Step 1 records (Relative finite-parameter avoidance of reducible connections).

[F3]

$π_{j} (X, x)$ is represented by continuous maps $u : (I^{j}, \partial I^{j}) \to (X, x)$ , with the zero class exactly the maps based-homotopic to the constant map (Higher homotopy group by based cubes).

Proof

technique · affine fillings followed by relative avoidance

1.1A1F1F2givenconstruct

The affine space $A_{\infty}$ contains $A_{0}$ . Apply [F2] to the constant map from a one-point polyhedron with empty subpolyhedron and value $A_{0}$ . Its endpoint is an element of $A_{\infty}^{*}$ , proving nonemptiness.

1.2F1F2constructalgebra

Let $A, B \in A_{\infty}^{*}$ . The affine path $γ (t) = (1 - t) A + t B$ is continuous in the Fréchet topology and has irreducible endpoints. Apply [F2] with $K = I$ and $L = {0, 1}$ . The resulting endpoint path remains in $A_{\infty}^{*}$ and fixes both endpoints, so the locus is path-connected.

1.3F1F3givenalgebraconstruct

Fix $j \geq 1$ and a based representative $u : (I^{j}, \partial I^{j}) \to (A_{\infty}^{*}, A)$ . The quotient $I^{j} / \partial I^{j}$ is homeomorphic to $S^{j}$ , so regard $u$ as a continuous sphere map with value $A$ at the collapsed boundary point. Identify $D^{j + 1}$ with the cone on $S^{j}$ and define $F ([r, x]) = A_{0} + r (u (x) - A_{0})$ for $0 \leq r \leq 1$ . For each continuous section seminorm $p$ , compactness of $S^{j}$ gives $C_{p} = \sup_{x} p (u (x) - A_{0}) < \infty$ ; hence $p (F ([r, x]) - A_{0}) \leq r C_{p} \to 0$ as $r \to 0$ . This proves continuity at the cone apex in every defining seminorm, and away from the apex continuity follows from continuity of $u$ and affine operations. Thus $F : D^{j + 1} \to A_{\infty}$ is a continuous filling of $u$ .

2.1F2F3step 1.3constructalgebra∎

Apply [F2] to $F$ with $K = D^{j + 1}$ and $L = \partial D^{j + 1} = S^{j}$ . Its boundary is already irreducible, so the endpoint is a map $G : D^{j + 1} \to A_{\infty}^{*}$ agreeing with $u$ on the boundary. Choose the based point $x_{0} \in S^{j}$ with $u (x_{0}) = A$ . Since the disk is convex, $C_{t} (x) = (1 - t) x + t x_{0}$ remains in the disk and fixes $x_{0}$ . Then $G (C_{t} (x))$ is a homotopy from $G$ to the constant map at $A$ that fixes $x_{0}$ and remains in $A_{\infty}^{*}$ . Restricting it to $S^{j}$ gives a based nullhomotopy of $u$ ; by [F3], its class in $π_{j} (A_{\infty}^{*}, A)$ is zero. Since $j$ and $u$ were arbitrary, every stated homotopy group vanishes.

Weak contractibility of the irreducible connection locus

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources