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The framed unknot represents a generator of pi_3 of S^2

Example

Assume the Axiom of Choice (The Axiom of Choice), used by the numerable-bundle lifting theorem in step 1.2 and supplying the countable choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) inherited by the regular-preimage and collapse constructions and the Pontryagin–Thom correspondence. Let $h : S^{3} \subseteq C^{2} \to {C P}^{1} ≅ S^{2}$ , $h (z_{1}, z_{2}) = [z_{1} : z_{2}],$ be the Hopf map. The fibre over $y : = [1 : 0]$ is the standard unknot $U = {z \in S^{3} : z_{2} = 0} ≅ S^{1},$ and $y$ is a regular value of $h$ ; the differential of $h$ along $U$ frames the normal bundle of $U$ in $S^{3}$ , so for a positive basis $b$ of $T_{y} S^{2}$ the pair $(U, h_{*} b)$ is a framed regular preimage (Framed regular preimages of a map to a sphere). Then the Pontryagin-Thom map of $(U, h_{*} b)$ is homotopic to $h$ , and since the Hopf fibration's long exact sequence gives $h_{*} : π_{3} (S^{3}) \to π_{3} (S^{2})$ an isomorphism while $π_{3} (S^{3}) ≅ Z$ by degree, the class of the framed unknot is a generator of $π_{3} (S^{2}) ≅ Z$ .

Facts & Assumptions

Given: The Hopf map $h : S^{3} \to {C P}^{1} ≅ S^{2}$ , $h (z_{1}, z_{2}) = [z_{1} : z_{2}]$ , with $y = [1 : 0]$ , the fibre $U = h^{- 1} (y) = {z_{2} = 0} ≅ S^{1}$ , and a positive basis $b$ of $T_{y} S^{2}$ .

[F1]

The Hopf map is a smooth surjection, $y$ is a regular value, $U$ is a closed embedded circle, and $h_{*} b$ is the framing of $ν (U \subseteq S^{3})$ induced by the differential (Framed regular preimages of a map to a sphere).

[F2]

The Pontryagin-Thom map of a framed regular preimage is smoothly homotopic to the original map (The collapse of a regular preimage is homotopic to the original map).

[F3]

The complex Hopf map is a numerable principal circle bundle over ${C P}^{1}$ ; numerable fibre bundles are Hurewicz fibrations under AC (Numerable fiber bundles are hurewicz fibrations).

[F4]

A based Serre fibration has a long exact sequence of homotopy groups (Long exact sequence of homotopy groups of a fibration).

[F5]

Based maps $S^{k} \to S^{r}$ are nullhomotopic for $0 \leq k < r$ (Lower-dimensional sphere maps are based nullhomotopic), and degree is an isomorphism $π_{r} (S^{r}) \to Z$ (Based sphere maps are classified by degree).

[F6]

The universal cover of the circle is $R$ , so $π_{k} (S^{1}) = 0$ for $k \geq 2$ : a map $S^{k} \to S^{1}$ with $k \geq 2$ lifts along the covering projection because $S^{k}$ is simply connected, and $R$ is contractible. The unit-circle and quotient-circle models agree by $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle; the lift exists by Lifting criterion for maps from path-connected locally path-connected spaces because spheres are path-connected and locally path-connected and their fundamental group is trivial ( $R \to R / Z$ is a universal covering, $S^{n}$ is simply connected for every $n \geq 2$ , Covering homotopies lift by finite local strips).

[F7]

The Pontryagin-Thom correspondence identifies framed cobordism classes of closed framed $1$ -submanifolds of $S^{3}$ with $π_{3} (S^{2})$ (The Pontryagin-Thom correspondence in fixed codimension, with $n = 3$ , $k = 2$ ).

Verification

1.1F1F3givenconstructalgebra

In the affine chart $[1 : w]$ the target coordinate is $w = z_{2} / z_{1}$ . At $(z_{1}, 0) \in U$ , its transverse differential is $δ z_{2} \mapsto δ z_{2} / z_{1}$ , an invertible complex map, hence of real rank two. Thus $y$ is regular, and its fibre is exactly $U = {∣ z_{1} ∣ = 1, z_{2} = 0}$ . This circle bounds the hemisphere disk ${x_{4} = 0, x_{3} \geq 0}$ in $S^{3}$ , so it is the standard unknot. The target identification with $S^{2}$ is smooth: in the $w$ chart it is $(2 ℜ w, 2 ℑ w, 1 - ∣ w ∣^{2}) / (1 + ∣ w ∣^{2})$ , the inverse stereographic formula, with the analogous formula in the other chart. Both affine charts also give smooth bundle sections $(1, w) / \sqrt{1 + ∣ w ∣^{2}}$ and $(v, 1) / \sqrt{1 + ∣ v ∣^{2}}$ ; multiplying by $S^{1}$ gives the bundle charts. To supply numerability, let $a = ∣ z_{1} ∣^{2}$ on unit representatives, which is well defined on the base. Put $b_{1} = σ (4 a - 1)$ , $b_{2} = σ (3 - 4 a)$ and $λ_{i} = b_{i} / (b_{1} + b_{2})$ . Their denominator is positive for $0 \leq a \leq 1$ , their sum is one, and the supports lie in $a \geq 1 / 4 \subset {z_{1} \neq 0}$ and $a \leq 3 / 4 \subset {z_{2} \neq 0}$ . This is a supplied finite support-subordinate partition of unity. The regular-preimage definition gives the framing $h_{*} b$ .

1.2F3F4F5F6

(The Hopf map generates $π_{3} (S^{2})$ .) The Hopf map is a numerable circle bundle and hence a Hurewicz, in particular Serre, fibration by [F3]; its long exact sequence by [F4] contains $π_{3} (S^{1}) \to π_{3} (S^{3}) \overset{h_{*}}{\to} π_{3} (S^{2}) \to π_{2} (S^{1})$ . By [F6] the outer groups vanish ( $k = 3 \geq 2$ and $k - 1 = 2 \geq 2$ ), so $h_{*}$ is an isomorphism; by [F5] $π_{3} (S^{3}) ≅ Z$ . Hence $π_{3} (S^{2}) ≅ Z$ , generated by the class of $h$ .

2.1F2step 1.1

(Its Pontryagin-Thom map is the Hopf map up to homotopy.) By [F2] the Pontryagin-Thom map $f_{(U, h_{*} b)} : S^{3} \to S^{2}$ is smoothly homotopic to $h$ ; in particular their classes in $π_{3} (S^{2})$ agree.

3.1F1F2F3F7step 2.1step 1.2∎

(Conclusion.) The framed unknot's Pontryagin-Thom class is the class of $h$ by step 2.1, which generates $π_{3} (S^{2}) ≅ Z$ by step 1.2, and the correspondence of [F7] identifies framed cobordism classes of framed links in $S^{3}$ with $π_{3} (S^{2})$ ; so $(U, h_{*} b)$ represents a generator. Full AC supplies both the hypothesis of the numerable-bundle lifting theorem [F3] and the countable choice inherited by the regular-preimage and collapse constructions [F1, F2] and the correspondence [F7]; no Hopf invariant theory is developed.

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Sources

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002) (standard reference, not scraped)
John Milnor, Topology from the Differentiable Viewpoint (standard reference, not scraped)