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Low special-unitary homotopy inputs

Statement

For every $N \geq 2$ , $S U (N)$ is path connected, $π_{1} (S U (N)) = π_{2} (S U (N)) = 0$ , and $π_{3} (S U (N)) ≅ Z$ . In addition, $π_{4} (S U (2)) ≅ Z / 2$ is finite and $π_{5} (S U (2)) \neq 0$ . Here $S U (2) ≅ S^{3}$ by the map sending a unit quaternion to its defining special-unitary matrix.

Statement references

Hatcher, Algebraic Topology, Chapter 4

Hatcher, Algebraic Topology, Appendix

May, A Concise Course in Algebraic Topology

Facts & Assumptions

Given: The column fibrations of Special-unitary column fibration, the Hopf and suspension results below, and the imported sphere and fibration theorems. Work in ZF + AC because the numerable-bundle supplier uses AC.

[F1]

The column maps form Hurewicz fibrations $S U (N - 1) \to S U (N) \to S^{2 N - 1}$ . Special-unitary column fibration

[F2]

For a based Serre fibration, the homotopy groups and component sets of fiber, total space, and base form an exact sequence. Long exact sequence of homotopy groups of a fibration

[F3]

For $k < d$ , every based map $S^{k} \to S^{d}$ is nullhomotopic; degree identifies $π_{3} (S^{3})$ with $Z$ . Lower-dimensional sphere maps are based nullhomotopic, Based sphere maps are classified by degree

[F4]

For an $(n - 1)$ -connected based CW complex, the unreduced two-cone suspension map $E : π_{i} (X, x_{0}) \to π_{i + 1} (Σ_{t w o - c o n e} X, p_{-})$ is surjective at $i = 2 n - 1$ . At $X = S^{2}$ , $n = 2$ , and $i = 3$ , this is the two-cone map out of $π_{3} (S^{2})$ . Freudenthal suspension theorem

[F5]

For integers $k \geq 0$ and $n \geq 1$ , the published sphere-system bonding map $E : π_{n + k} (S^{n}) \to π_{n + k + 1} (S^{n + 1})$ is an isomorphism when $n > k + 1$ and is surjective when $n = k + 1$ . Its proof identifies this map with reduced suspension by collapsing the contractible basepoint track in the two-cone suspension; at $n = 2$ , $k = 1$ , this is precisely the reduced map $π_{3} (S^{2}) \to π_{4} (S^{3})$ . Freudenthal identifies the eventual suspension system for spheres

[F6]

The Hopf generator $h$ generates $π_{3} (S^{2})$ , $2 [Σ h] = 0$ , and $(Σ h) \circ (Σ^{2} h) \neq 0$ . Hopf generator and mapping-cone coefficient, Folded product attachment has Hopf invariant two and null suspension, Nonvanishing of consecutive suspended Hopf composites by Adem relation

Proof

1.1givenF3algebra

The unit-quaternion parametrization $q = a + b i + c j + d k \mapsto (\begin{array}{cc} a + b i & c + d i \\ - c + d i & a - b i \end{array})$ is a continuous bijection from compact $S^{3}$ to Hausdorff $S U (2)$ ; direct multiplication shows it is unitary with determinant one, and every matrix in $S U (2)$ has this form. It is therefore a homeomorphism. By [F3], $π_{1} (S^{3}) = π_{2} (S^{3}) = 0$ and $π_{3} (S^{3}) = Z$ .

2.1F4F5F6step 1.1algebra

By [F4], the two-cone Freudenthal suspension map on $π_{3} (S^{2})$ is surjective at the boundary index. The comparison in [F5], with $n = 2$ and $k = 1$ , identifies this map with reduced suspension, so $E [h] = [Σ h] = η_{3}$ generates the quotient $π_{4} (S^{3})$ . The order relation $2 η_{3} = 0$ and the nonzero composite $η_{3} \circ η_{4}$ are supplied by [F6]. The latter implies $η_{3} \neq 0$ , since a composite with a null factor is null. Thus the cyclic quotient generated by $η_{3}$ has exactly two elements, proving $π_{4} (S U (2)) ≅ Z / 2$ ; the same nonzero composite lies in $π_{5} (S^{3})$ , so that group is nonzero.

3.1F1F2F3step 1.1algebra∎

Induct on $N \geq 3$ . The base sphere $S^{2 N - 1}$ has dimension at least five, so [F3] makes its homotopy groups in degrees $1, 2, 3, 4$ zero. Exactness [F2] at degrees one and two therefore gives $π_{1} (S U (N - 1)) ≅ π_{1} (S U (N))$ and $π_{2} (S U (N - 1)) ≅ π_{2} (S U (N))$ ; at degree three it gives $π_{3} (S U (N - 1)) ≅ π_{3} (S U (N))$ . The component exact sequence and the connected fiber and base show $S U (N)$ is path connected: the map on components is onto, and its kernel is the image of the one-point component set of the fiber. Starting from [step 1.1], induction gives all three displayed low groups and connectedness for every $N$ .

Low special-unitary homotopy inputs

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources