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Nontrivial homotopy of reduced sphere gauge groups

Statement

Let $r \in {3, 4}$ , $N \geq 2$ , and $P \to S^{r}$ be any smooth principal $S U (N)$ bundle. Write $G (P) : = G_{\infty} (P)$ for its smooth gauge group with the finite-atlas $C^{\infty}$ section topology and $\overline{G} (P) : = {\overline{G}}_{\infty} (P) = G_{\infty} (P) / Z_{N}$ for the quotient by constant central gauge sections. Then either $\overline{G} (P)$ has more than one path component or at least one of its based homotopy groups $π_{j} (\overline{G} (P), e)$ , $j \geq 1$ , is nonzero. More precisely, for $r = 3$ it has infinitely many path components, and for $r = 4$ its fundamental group is nontrivial.

Statement references

May, A Concise Course in Algebraic Topology, Chapters 7 and 9

Hatcher, Algebraic Topology, Chapter 4

Facts & Assumptions

Given: The sphere bundle, chosen evaluation frame, smooth section topologies, and finite central quotient of the Statement. The actual configuration carrier is present, but its current hash-bound Step 1 record is ready according to loadStep1Record and step1Decision; this is authoring readiness, not independent approval.

[F1]

The actual configuration definition fixes compact connected oriented boundaryless Riemannian $M$ of dimension $3$ or $4$ , $P$ , its frame, the smooth gauge group $G_{\infty}$ with finite-atlas $C^{\infty}$ section topology, the based subgroup, and ${\overline{G}}_{\infty} = G_{\infty} / Z_{N}$ ; it also defines the Sobolev groups and quotient topologies. Its carrier was inspected in full, but its current hash-bound Step 1 record is ready according to loadStep1Record and step1Decision; this is authoring readiness, not independent approval. Smooth and Sobolev configuration and reduced gauge spaces

[F2]

Evaluation is a numerable fibration $G_{m} (P) \to G (P) \to S U (N)$ and $G (P) \to \overline{G} (P)$ is a finite covering. Evaluation and finite-center fibrations of sphere gauge groups

[F3]

For every bundle sector, $π_{0} (G_{m} (P)) ≅ π_{r} (S U (N))$ and $π_{j} (G_{m} (P)) ≅ π_{j + r} (S U (N))$ for $j \geq 1$ . Based gauge-group homotopy via hemisphere clutching

[F4]

$S U (N)$ is path connected, $π_{1} (S U (N)) = π_{2} (S U (N)) = 0$ , and $π_{3} (S U (N)) ≅ Z$ . Low special-unitary homotopy inputs

[F5]

$π_{5} (S U (N)) \neq 0$ for all $N \geq 2$ . Nonvanishing of the fifth special-unitary homotopy group

[F6]

The fibration LES is exact on homotopy groups and component sets; a covering induces an injection on fundamental groups. Long exact sequence of homotopy groups of a fibration, A covering map induces an injective homomorphism on fundamental groups

[F7]

Paths lift to a covering once their initial lift is fixed. Existence and uniqueness of path lifts through a covering map

[A1]

Use the actual smooth topology and constant central subgroup fixed by [F1]. No irreducible-configuration claim is needed for the homotopy argument.

Proof

1.1F1F2F3F4F6F7

Suppose $r = 3$ . By [F3, F4], $π_{0} (G_{m} (P)) ≅ π_{3} (S U (N)) ≅ Z$ . The evaluation LES [F2, F6] has the component segment $π_{1} (S U (N)) ⟶ π_{0} (G_{m} (P)) ⟶ π_{0} (G (P)) ⟶ π_{0} (S U (N)) .$ The outer terms are one-point sets by [F4]. The induced map on components is surjective by exactness, and its fibers are the orbits of $π_{1} (S U (N))$ on $π_{0} (G_{m} (P))$ in the action clause of [F6]. That acting group is trivial, so each fiber is a singleton and the middle map is a bijection. Thus $G (P)$ has infinitely many components. Write $q : G (P) \to \overline{G} (P)$ for the quotient covering. Translation $u \mapsto u z$ by any $z \in Z_{N}$ is a homeomorphism and therefore acts on path components. If $q (x)$ and $q (y)$ are joined by a path $α$ , lift $α$ starting at $x$ by [F7]. Its endpoint lies in $q^{- 1} (q (y)) = y Z_{N}$ , so $x$ is joined upstairs to $y z$ for some $z \in Z_{N}$ . Conversely, any path from $x$ to $y z$ projects to a path from $q (x)$ to $q (y)$ . Since $q$ is surjective, its induced map on component sets is surjective, and these two implications identify its fibers exactly with the $Z_{N}$ -orbits. Each orbit has at most $N$ elements; finitely many quotient components would therefore make the upstairs component set finite. Thus infinitely many components remain downstairs.

2.1F2F3F4F5F6algebra∎

Suppose $r = 4$ . By [F3] and [F5], $π_{1} (G_{m} (P)) ≅ π_{5} (S U (N)) \neq 0$ . In the evaluation LES, the map $π_{1} (G_{m} (P)) \to π_{1} (G (P))$ has preceding term $π_{2} (S U (N)) = 0$ by [F4]; exactness makes it injective. The finite covering in [F2] induces an injection $π_{1} (G (P)) ↪ π_{1} (\overline{G} (P))$ by [F6]. Hence $π_{1} (\overline{G} (P))$ contains a nonzero subgroup. The two cases establish the statement for every $r, N, P$ .

Nontrivial homotopy of reduced sphere gauge groups

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources