1.2 Computation of Homotopy groups

We have the S¹-fibration (Hopf fibration) S^{2n + 3}$\to$$\mbox{$\bf P$}$^{n + 1}. Let $\widetilde{Q}$$\to$Q be the restriction of this to Q.

Scholium 3   $\widetilde{Q}$ is the total space of the unit sphere bundle of the tangent bundle of S^{n + 1}. Hence we have a fibration

Sⁿ$$\to$$$$\widetilde{Q}$$$$\to$$S^{n + 1}.$$\eqno$$(*)

Proof:     If (z₀ : ^... : z_n) are homogeneous coordinates on $\mbox{$\bf P$}$^{n + 1}, u_i = $\Re$(z_i) and v_i = $\Im$(z_i), then we may take

S^{2n + 3} = {(u_i, v_i) | $$\sum$$u_i² + $$\sum$$v_i² = 2}.

We may assume that Q is defined in these coordinates by the equation $\sum$z_i² = 0, so that

$$\widetilde{Q}$$ = {(u_i, v_i) | $$\sum$$u_i² = $$\sum$$v_i² = 1 and $$\sum$$u_iv_i = 0}.

Hence, the projection to the u_i's is a fibration of the required sort. $\Box$
For any subset A $\subset$ Q, let $\widetilde{A}$ denote its inverse image in $\widetilde{Q}$, so that there is an induced S¹-fibration $\widetilde{A}$$\to$A. For any abelian group G let ₂G denote its 2-torsion subgroup and let G/2 = G $\otimes$ $\mbox{$\bf Z$}$/2$\mbox{$\bf Z$}$.

Lemma 4  

(iii)

If n is odd, then for i $\leq$ 2n - 1, we have a short exact sequence

0$$\to$$$$\pi_{i}^{}$$(Sⁿ)/2$$\to$$$$\pi_{i}^{}$$($$\widetilde{Q}$$)$$\to$$₂$$\pi_{i}^{}$$(S^{n + 1})$$\to$$0.

(iv)

If n is even, then we have a split exact sequence for each i

0$$\to$$$$\pi_{i}^{}$$(Sⁿ)$$\to$$$$\pi_{i}^{}$$($$\widetilde{Q}$$)$$\to$$$$\pi_{i}^{}$$(S^{n + 1})$$\to$$0.

Proof:     In the long exact sequence

^... $$\to$$$$\pi_{i}^{}$$(Sⁿ)$$\to$$$$\pi_{i}^{}$$($$\widetilde{Q}$$)$$\to$$$$\pi_{i}^{}$$(S^{n + 1})$$\;\stackrel{\partial_i}{\rightarrow}\;$$$$\pi_{i-1}^{}$$(Sⁿ)$$\to$$ ^...

the boundary maps $\partial_{i}^{}$ are the maps on $\pi_{i}^{}$ induced by a map (well defined upto homotopy) $\Delta$ : $\Omega$S^{n + 1}$\to$Sⁿ coming from the fibration (*). Suppose s : Sⁿ$\to$$\Omega$S^{n + 1} = $\Omega$$\Sigma$Sⁿ is the map inducing the suspension homomorphisms $\Sigma_{i}^{}$ : $\pi_{i}^{}$(Sⁿ)$\to$$\pi_{i+1}^{}$(S^{n + 1}). Since (*) is the spherical fibration associated to the tangent bundle of S^{n + 1}, it is well known (see [W] IV, (10.4)) that if $\theta$ = $\Delta$os,

[$$\theta$$] = 1 + (- 1)^{n + 1} $$\in$$ $$\pi_{n}^{}$$(Sⁿ)

where 1 $\in$ $\pi_{n}^{}$(Sⁿ) is the standard generator. From the Freudenthal Suspension theorem, $\Sigma_{i}^{}$ is an isomorphism for i $\leq$ 2n - 2, and $\Sigma_{2n-1}^{}$ is a surjection. We shall often make use of the following well known result (see [W] XI, (1.11), (1.12), (1.16)).

Scholium 5   Let f : Sⁿ$\to$Sⁿ be any continuous map of degree d, where n > 1. Then the induced map

f_* : $$\pi_{i}^{}$$(Sⁿ)$$\to$$$$\pi_{i}^{}$$(Sⁿ)

is

(v)

multiplication by d if i < 2n - 1

(vi)

multiplication by d on the torsion subgroup, for i = 2n - 1 (and in particular on $\pi_{2n-1}^{}$, if n is odd)

(vii)

multiplication by d² on $\pi_{2n-1}^{}$(Sⁿ) $\otimes$ $\mbox{$\bf Q$}$.

$\Box$

If n is even, then [$\theta$] = 0 so that $\theta_{*}^{}$ = 0. Hence, $\partial_{n+1}^{}$ vanishes. Thus (*) has a homotopy section and the long exact sequence for this fibration splits into short exact sequences as asserted in (ii). If n is odd, Scholium 5 implies that $\theta_{*}^{}$ acts as multiplication by 2 on $\pi_{i}^{}$(Sⁿ) in the range i $\leq$ 2n - 1. This proves (i). $\Box$