next up previous
Next: 2 The odd dimensional Up: 1 Preliminaries Previous: 1.1 Cell structure

1.2 Computation of Homotopy groups

We have the S1-fibration (Hopf fibration) S2n + 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 Sn + 1. Hence we have a fibration

Sn$\displaystyle \to$$\displaystyle \widetilde{Q}$$\displaystyle \to$Sn + 1.$\displaystyle \eqno$(*)

Proof:     If (z0 : ... : zn) are homogeneous coordinates on $ \mbox{$\bf P$}$n + 1, ui = $ \Re$(zi) and vi = $ \Im$(zi), then we may take

S2n + 3 = {(ui, vi) | $\displaystyle \sum$ui2 + $\displaystyle \sum$vi2 = 2}.

We may assume that Q is defined in these coordinates by the equation $ \sum$zi2 = 0, so that

$\displaystyle \widetilde{Q}$ = {(ui, vi) | $\displaystyle \sum$ui2 = $\displaystyle \sum$vi2 = 1 and $\displaystyle \sum$uivi = 0}.

Hence, the projection to the ui'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$\displaystyle \to$$\displaystyle \pi_{i}^{}$(Sn)/2$\displaystyle \to$$\displaystyle \pi_{i}^{}$($\displaystyle \widetilde{Q}$)$\displaystyle \to$2$\displaystyle \pi_{i}^{}$(Sn + 1)$\displaystyle \to$0.

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

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

Proof:     In the long exact sequence

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

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

[$\displaystyle \theta$] = 1 + (- 1)n + 1 $\displaystyle \in$ $\displaystyle \pi_{n}^{}$(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* : $\displaystyle \pi_{i}^{}$(Sn)$\displaystyle \to$$\displaystyle \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$

next up previous
Next: 2 The odd dimensional Up: 1 Preliminaries Previous: 1.1 Cell structure
Kapil Hari Paranjape 2002-11-21