≡1 mod 2 and Theorem 3.13 gives us that Sq2bαn−1−2b = αn−1.
To prove the second statement, suppose j = 2s(2t+ 1) where 0≤s < b. Then we know the coefficient of 2s in the binary expansion ofj is 1. (Why? If j=Pm
i=0ai2i is
the binary expansion ofj, thenai= 0 for all i < s. It follows thatas= 0 implies 2s+1
dividesj. Soas must be 1.) Since
n−1 = 2b−1 + 2b+1a=
b−1
X
i=0
2i+ 2b+1a,
the coefficient of 2s in the binary expansion n−1 is 1. Thus, the coefficient of 2s in the binary expansion of n−j−1 is 0. Lemma 3.14 implies n−jj−1
≡ 0 mod 2 and Theorem 3.13 gives the required result.
3.3
Proving the special case
The following is a simplified version of Steenrod and Whitehead’s original proof, adapted from [9, page 494].
Proof. When a= 0, the result is trivially true. In this case, n= 2b and ρ(n) = 2b. It is not possible to have 2b orthogonal vectors tangent to a point inS2b−1 ⊂R2b – there simply are not enough dimensions. Therefore, we can assumea >0. Letk=ρ(n) + 1. Suppose we have k−1 vector fields onSn−1. By Lemma 1.40, equivalently suppose we have section f :Sn−1→Vn,k of the bundlep:Vn,k→Sn−1. Then
Hn−1(Sn−1;Z2)
p∗
−→Hn−1(Vn,k;Z2)
f∗
−→Hn−1(Sn−1;Z2)
is the identity. It follows that f∗ is surjective. We can always deform f to be a cellular map and then its image will be contained in the (n−1)-skeleton of Vn,k.
We know that 2k−1 = 2ρ(n) + 1 = 2b+1+ 1≤n. So Proposition 1.41 tells us that the (n−1)-skeleton of Vn,k is RPnn−−1k. Thus, by deforming f if necessary, we obtain a
mapg:Sn−1 →RPnn−−1k with Sn−1 −→g RPnn−−1k p| RPn−1 n−k −−−−−→Sn−1 the identity. Theng∗ is surjective on Hn−1(−;Z2). Since
Hn−1(RPnn−−1k;Z2)∼=H
n−1(Sn−1;
Z2)∼=Z2, it follows that g∗ is an isomorphism onHn−1.
Now, the inclusion i : RPnn−−1k → RPn−1 induces an isomorphism on Hn−1(−;Z2). Thus,g∗i∗ :Hn−1(RPn−1;Z2)
∼ =
−→Hn−1(Sn−1;Z2) is an isomorphism. By Theorem 3.15, we know that
is non-zero. But
Sqk−1 :Hn−k(Sn−1;Z2)→Hn−1(Sn−1;Z2)
is obviously zero. We then have a contradiction by using the naturality of Steenrod squares. Specifically, we know thatg∗i∗Sqk−1αn−k =g∗i∗αn−1 6= 0, whereαn−k is the generator of Hn−k(RPn−1;Z2). But Sqk−1g∗i∗αn−k must be 0.
If we replacedk= 2c+ 1 withj≤2c, then
Sqj−1 :Hn−j(RPn−1;Z2)→Hn−1(RPn−1;Z2), αn−j 7→0,
by Theorem 3.15 and so we would no longer gain a contradiction. This shows that the upper bound of 2c on the number of vector fields is the best possible provided by this argument. So this argument can only prove that ρ(n) is the upper bound whend= 0. We need more sophisticated tools (i.e.K-theory) to prove the general case.
Chapter 4
The Stable Homotopy Category
We are now at the point where we need to develop some significant amount of the- ory before we can make more progress on our vector fields problem. The next three chapters—on the stable homotopy category, duality and spectral sequences—may seem to be a detour, but they have important applications to our task, as well as many other areas of algebraic topology.
In this chapter, we will introduce the stable homotopy category. There are a number of constructions of the stable homotopy category. All these constructions first build a category of objects, called spectra. Then they replace the morphisms in this category with some notion of homotopy classes of maps. The resulting category is called the stable homotopy categoryHo(Spectra). These constructions are analogous to defining the category CW of CW complexes and then using this to construct the category
Ho(CW) of CW complexes with homotopy classes of maps.
All of these constructions result in equivalent categories. So we can talk about the
stable homotopy category.
We will primarily present the historical construction, given first by Boardman [23]†. While the stable homotopy category enjoys a symmetric smash product, Boardman’s category of spectra does not. It was only in 1997 when a category of spectra with a symmetric smash product was discovered by Elmendorf, Kriz, Mandell and May [7]. We will briefly present the category of symmetric spectra, first given in [12], to highlight the advantages of these modern spectra.
As indicated by the name, the stable homotopy category is the natural environment for stable homotopy theory. One advantage is that the suspension functor
Σ :Ho(Spectra)→Ho(Spectra)
is an equivalence, as we shall prove later. So we can invert suspensions and consequently many phenomenon are stable under suspension inHo(Spectra). Suspension on topo- logical spaces is not so well behaved. In this sense, Ho(Spectra) is the ‘stabilisation’ of the classical homotopy categoryHo(Top).
†
However, Adams attributes the notion of spectra to Lima (in [17]) and G. W. Whitehead [2, p. 131]
In this chapter, we assume every space is locally compact Hausdorff.