In this paragraph, we will briefly recall some topological properties of spheres and their loop spaces that will be used in the proofs of Theorem 4.2 and Theorem 4.3.
If k is odd, let Lk denote the k-dimensional sphereSk. The cohomology
ring is the exterior algebra
H∗(Lk;Z)∼= ΛZ[α]
with α∈Hk(Lk;Z) a generator. Therefore, by the K¨unneth formula H∗(Lbk;Z)∼= ΛZ[α1, . . . , αb]
withαi of degree k. Furthermore, we will always use the CW decomposition
of Sk consisting of one 0-cell and one k-cell.
It is known by work of Serre (see [Ser51]) that the homotopy groups
πm(Sk) for m > k are finite (recall that k is odd). Sullivan showed in
[Sul74] that the selfmap Sk → Sk of degree d induces a homomorphism
πm(Sk) → πm(Sk) which is nilpotent on d-torsion. Therefore, there exists
a nonzero degree selfmap of Sk that induces the trivial homomorphism on πm(Sk).
Finally, note that the map Sk→Sk of degree two multiplies each homol-
ogy class in Lbk of positive dimension by some power of two. In particular, the induced homomorphism on homology withZ2 coefficients is trivial.
If k is even, let Lk be a CW complex that is homotopy equivalent to the
based loop space ΩSk+1 of the (k+ 1)-dimensional sphere. More precisely, letLk be the James reduced product J(Sk) (see for example [Hat02], pages
224–225 and section 4.J). The CW structure onLk consists of one cell in each
dimension divisible by k, and the cohomology ring is the divided polynomial algebra
H∗(Lk;Z)∼= ΓZ[α]
with α ∈ Hk(Lk;Z). (Recall that this is almost a polynomial algebra. In fact, the generator in degree kp equals αp/p!. Using real coefficients, the
cohomology ring is a polynomial algebra.) By the K¨unneth formula
H∗(Lbk;Z)∼= ΓZ[α1, . . . , αb],
where the classes αi are of degree k.
Since πm(ΩSk+1) =πm+1(Sk+1), the homotopy groups πm(Lk) are finite
for m > k (recall that k is even now). The group structure in πm(Lk)
coincides with the one coming from loop multiplication, hence one can easily construct selfmaps Lk → Lk that induce multiplication by some positive
integer onHk(L
k;Z) and trivial homomorphisms on πm(Lk).
The map ΩSk+1 → ΩSk+1 that assigns to each loop its double induces multiplication by two inπk(Lbk) and therefore also ink-dimensional homology Hk(Lbk;Z) and cohomology H
k(Lb
k;Z). Hence, it induces multiplication by
some power of two in everyH`(Lb
k;Z) with` >0 and thus in everyH`(Lbk;Z),
as well. Consequently, the induced homomorphism on homology with Z2 coefficients is trivial.
Summary. The CW complexes Lb
k have cells only in dimensions divisible by k, their real cohomology rings are generated by elements of degree k, and there exist selfmaps hm : Lbk →Lbk for all m > k that induce multiplication by some positive integer onHk(Lb
k;Z) (and therefore also onHk(Lbk;Z)) and
vanish on πm(Lbk). Moreover, there exists a selfmap Lbk → Lbk that induces
the zero homomorphism in homology of nonzero dimension with coefficients inZ2 but that is bijective on Hk(Lbk;R).
Using these properties of Lk, we are able to prove the following lemma. It stems from [KatSu99], sections 4 and 10.
Lemma 4.11. Let X be a connected CW complex of dimension n. Let 1 ≤
there is a map f :X →Lb
k that induces an isomorphism
f∗ :Hk(X;R) ∼ =
−→Hk(Lbk;R).
In fact, f can be chosen such that the induced map on the integral lattices corresponds to multiplication by some positive integer.
Proof. The case k = 1 is easy because Lb1 is just theb-dimensional torusTb, which is K(Zb,1). The canonical epimorphism
π1(X)H1(X;Z)H1(X;Z)R∼=Z b
is induced by the so-called Jacobi map f :X →Tb. The induced homomor-
phism f∗ : H1(X;R)→ H1(Tb;R) is consequently an isomorphism, which is moreover an isomorphism of the integral lattices.
Now, let 2 ≤ k ≤ n−1. Choose a CW decomposition of K(Zb, k) such
that the (k+ 1)-skeleton isWb
Sk, the wedge sum ofbspheres of dimensionk.
As shown in the introductory paragraph of this chapter, there is a mapX →
K(Zb, k) that induces an isomorphism on the integral lattices of homology in
dimension k. By cellular approximation, this gives a map X(k+1) → Wb
Sk.
Note that the (k+ 1)-skeleton of Lbk is also Wb
Sk. Thus, we have a map
f(k+1):X(k+1) →Lbk
that induces an isomorphism on the integral lattices of homology in degree
k.
Let hk+1 : Lbk → Lbk be as in the summary above, i. e. it induces the
trivial homomorphism on the (k+ 1)-dimensional homotopy group and mul- tiplication by some positive integer on real homology of degree k. Then the composition hk+1 ◦ f(k+1) : X(k+1) → Lbk extends over X(k+2) since it
is zero on the (k + 1)-dimensional homotopy groups. Call this extension
f(k+2) :X(k+2) →Lb
k. Repeating this process finally gives a map f :X →Lbk
for which the induced monomorphismHk(X;Z)R,→Hk(Lbk;Z)Rcorresponds to multiplication by some positive integer.