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 + 3n + 1. Let Q be the restriction of this to Q.

Scholium 3   is the total space of the unit sphere bundle of the tangent bundle of Sn + 1. Hence we have a fibration

SnSn + 1.(*)

Proof:     If (z0 : ... : zn) are homogeneous coordinates on n + 1, ui = (zi) and vi = (zi), then we may take

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

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

= {(ui, vi) | ui2 = vi2 = 1 and uivi = 0}.

Hence, the projection to the ui's is a fibration of the required sort.
For any subset A Q, let denote its inverse image in , so that there is an induced S1-fibration A. For any abelian group G let 2G denote its 2-torsion subgroup and let G/2 = G /2.

Lemma 4
(iii)
If n is odd, then for i 2n - 1, we have a short exact sequence

0(Sn)/2()2(Sn + 1)0.

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

0(Sn)()(Sn + 1)0.

Proof:     In the long exact sequence

... (Sn)()(Sn + 1)(Sn) ...

the boundary maps are the maps on induced by a map (well defined upto homotopy) : Sn + 1Sn coming from the fibration (*). Suppose s : SnSn + 1 = Sn is the map inducing the suspension homomorphisms : (Sn)(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 = os,

[] = 1 + (- 1)n + 1 (Sn)

where 1 (Sn) is the standard generator. From the Freudenthal Suspension theorem, is an isomorphism for i 2n - 2, and 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 : SnSn be any continuous map of degree d, where n > 1. Then the induced map

f* : (Sn)(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 , if n is odd)
(vii)
multiplication by d2 on (Sn) .

If n is even, then [] = 0 so that = 0. Hence, 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 acts as multiplication by 2 on (Sn) in the range i 2n - 1. This proves (i).

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