Localization genus

Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.


Introduction
Classically the genus of a nilpotent space X of finite type, as introduced by Mislin in [16], consists of all homotopy types of nilpotent spaces Y of finite type such that the localizations Y (p) and X (p) coincide at any prime p. That is, spaces in the same genus as X cannot be distinguished from X if one looks at them through the eyes of p-localization. Another equivalent definition can be given in terms of rationalization and p-completion, [21].
We introduce in this article the notion of localization genus. A localization functor L in the category of spaces (or simplicial sets), as introduced by Bousfield, [3], Farjoun, [7], is a homotopy functor equipped with a natural transformation η from the identity which is idempotent up to homotopy. The main point to study such functors is that it subsumes the notions of localization at a prime or a set of primes (e. g. rationalization) and p-completion, but also Postnikov sections, Quillen's plus-construction, and other nullification or periodization functors such as P BZ/p , which plays a central role in the Sullivan conjecture, [15]. We writeLX for the homotopy fiber of the natural map η X : X → LX. We define thus two genus sets associated to L for any simply connected CW-space X of finite type.
(1) The extended L genus set for X is the setḠ L (X) = {Y | LY ≃ LX,LY ≃LX} of homotopy types Y of CW-spaces such that LY = LX andLY =LX.
(2) The L-genus set for X is the subset G L (X) ofḠ L (X) represented by CW-spaces of finite type. Our definition is motivated by the classical definition of the (completion) genus set, [16], [21,Definition 3.2], and the extended (completion) genus set studied by McGibbon in [13]. We show in fact in Proposition 1.3 that when L is rationalization, one gets back these classical notions. To illustrate our point of view we go through the computation of the extended rationalization genus G(S n ) of an odd sphere, Theorem 2.2, and we characterize in Corollary 2.4 those elements inḠ(S n ) corresponding to elements in the extended genus of the abelian group of integers, as studied by Hilton in [10].
To tackle technically harder problems we rely on Dywer and Kan's classifiyng space for towers of fibrations, [6], a tool which has proven to be handy in similar situations, [17]. This allows us in particular to do explicit computations of Postnikov genus sets for spheres and complex projective spaces.
Theorem 6.6. The extended Postnikov genus set G [n] (S n ) of homotopy types of spaces Y such that Y [n] ≃ K(Z, n) and Y n ≃ S n n is uncountable, in bijection with p N + , where the product is taken over all primes.
We also present in Section 4 a computation related to Neisendorfer's functor, [18], and the Sullivan conjecture. The localization genus computations show combined features of the space one focuses on and the chosen localization functor. The notion of genus quantifies in which sense it is (not) sufficient to consider a given space locally, through the eyes of a localization functor L and the associated fiberL.
Acknowledgements. This project started during a visit of the first author at the Universitat Autònoma de Barcelona in 2008 and was rebooted during a visit of the second author at the Centre for Symmetry and Deformation in Copenhagen seven years later.

Genus and extended genus
Let L be a homotopical localization functor, i.e. a coaugmented and idempotent homotopy functor in the category of spaces. It is sometimes more convenient to work in the Quillen equivalent category of simplicial sets, in particular when one needs models for mapping spaces. We will clearly say so when we do so. In practice localization functors arise as follows. To any map f one associates a functor L f which inverts f in a universal way, [7] and [3]. Clever choices for the map yield homological localization, localization at a set of primes such as rationalization, Quillen's plus construction, Postnikov sections, etc.
The homotopy fiber of the coaugmentation X → LX is LX. Definition 1.1. Let L be a localization functor and X a simply connected CW-complex of finite type.
• The extended-L genus set for X is the set of homotopy types Y of CW-spaces such that LY = LX and LY = LX.
• The L-genus set for X is the subset G L (X) of G L (X) represented by CW-complexes of finite type.
The reason for this "generic" terminology comes from the relationship with the classical notion of genus. Recall that Mislin's definition, [16], is given in terms of localization at primes: Two spaces X and Y belong to the same genus set if their localizations X (p) and Y (p) are homotopy equivalent at every prime p. This is a stronger requirement than merely asking for equivalent p-completions X ∧ p and Y ∧ p , for any prime p, and equivalent rationalizations X 0 and Y 0 , as shown for example by Belfi and Wilkerson in [2]. We will focus on the completion genus set as in [21,Definition 3.5] and the extended completion genus set, [13]. We denote by X ∧ the product of all p-completions. Definition 1.2. Let X be a simply connected CW-complex of finite type.
• The extended genus set of X is the set G(X) of homotopy types of CW-complexes Y such that Y ∧ = X ∧ and Y 0 = X 0 .
• The genus set of X is the subset G(X) of G(X) represented by CW-complexes Y of finite type.
We show now that the classical completion genus coincides with our localization genus, when the chosen localization functor L is rationalization. Since we restrict our attention to simply connected spaces here, one can choose the map f to be the wedge of degree p maps on the 2-sphere, taken over all primes p. Then the Bousfield localization functor L f is rationalization on simply connected spaces. When LX = X 0 is rationalization, we write X τ for LX, the fibre of the rationalization map X → X 0 and call it the torsion space of X. Proposition 1.3. Let X be a simply connected CW-complex of finite type. The extended rationalization-genus set is the extended genus set G(X) and the rationalization-genus set G 0 (X) coincides with the classical genus set G(X).
Let us finally remark that all spaces in G(X) are finite complexes when X is a finite complex.
This comes from the fact that, when X is of finite type, the integral homology groups of any space in the genus set of X are those of X. There is thus a Moore-Postnikov decomposition of such a space as successive homotopy cofibers of maps between (finite) Moore spaces, see for example [9, Chapter 8].

The extended rationalization genus of an odd sphere
In this section we turn our attention to a concrete example and propose an explicit computation of the rationalization-genus for odd spheres. Let n be an odd natural number. The extended rationalization genus set of the odd-dimensional sphere S n is according to [13, Theorem 3] an uncountable set. We offer an explicit description of this extended genus set and identify the elements known as pseudo-spheres. For this we will need some elementary abelian group theory.
Any torsion free abelian group A of rank one can be seen, up to isomorphism, as a subgroup of Q containing Z. For each prime p, let k p (A) = max{r ≥ 0 | 1 ∈ p r A} denote the height of 1 at p.
The height sequence of A is the sequence Π(A) = (k p (A)) p of non-negative (or infinite) integers.
Two sequences (k p ) and (m p ) are similar if the sum of the differences |k p − m p | is finite. This means that the sequences differ in only a finite number of primes, and have ∞ in the same coordinate.
A type is a similarity class of sequences. As explained in [8], or [19,Theorem 10.47], isomorphism types of torsion free abelian groups of rank one are in bijection with types.
We start now with the fibration S n τ → S n → (S n ) 0 and will use that, since n is odd, (S n ) 0 ≃ K(Q, n) ≃ M (Q, n) and S n τ ≃ M (Q/Z, n − 1). For any space Y in the extended rationalization genus of S n we have a fibration sequence M (Q/Z, n − 1) → Y → K(Q, n). The homotopy long exact sequence yields an exact sequence The idea is that the connecting homomorphism ∂ determines the homotopy type of Y . Proof. We define two maps. The first α : Hom(Q, Q/Z) → {A | A torsion free abelian of rank one} sends a homomorphism ∂ to its kernel. The second one we call β. Given a subgroup A of Q, let J be the subset of all primes consisting of those primes p for which n p = ∞. Then the quotient Q/A is isomorphic to ⊕ p∈J Z p ∞ and β sends A to the composite Clearly α•β sends a torsion free abelian group of rank one A to a subgroup of Q which is isomorphic to A. Moreover, given a homomorphism ∂ : Q → Q/Z, the image β(Ker ∂) coincides with ∂ up to an isomorphism of Q (corresponding to the choice of an inclusion α(∂) ⊂ Q) and an isomorphism of Q/Z (corresponding to the choice of an isomorphism Q/ Ker ∂ ∼ = ⊕ p∈J Z p ∞ ). This proves the lemma.
We proceed now with the construction for any homomorphism ∂ : in the extended genus G(S n ) realizing this homomorphism as connecting map in the homotopy and so there exists up to homotopy a unique map We define Y (∂) to be the homotopy cofiber of ∆. We have therefore a cofibration sequence which is seen to be a fibration sequence as well, for example by an elementary Serre spectral sequence argument (a complete characterization of such sequences has been obtained by Alonso, [1], see also Wojtkowiak's [22]).
Theorem 2.2. The extended genus set G(S n ) is in bijection with the set of isomorphism classes of torsion free abelian groups of rank one.
Proof. Given a torsion free abelian group A of rank one we get a homomorphism ∂ : Q → Q/Z from Lemma 2.1 and construct as above the space Y (∂). It realizes ∂ as connecting homomorphism in the homotopy long exact sequence, and its kernel is a group isomorphic to A. To show that we have indeed a bijection we must prove that the connecting homomorphism determines the homotopy type of Y ∈ G(S n ). Let ∂ be the connecting homomorphism for such a space Y and let us compare We have thus constructed a map Y (∂) → Y which induces equivalences on rationalizations and torsion spaces. It is hence an equivalence as well.
In the final part of the section we restrict our attention to the (n − 1)-connected members of the extended genus set of an odd sphere. We begin with a review of Hilton's investigations of the extended genus set of Z and groups of pseudo-integers [10, 11]. Since a group of pseudo-integers is a torsion free abelian group of rank one, in the terminology introduced at the beginning of the section, it is characterized by its type, which consists in only finite integers k p . This subset of torsion free abelian group of rank one has been studied by Hilton According to [10, Corollary 2.5], the extended genus set G(Z) of Z, consisting of isomorphism classes of (not necessarily finitely generated) abelian groups H such that H ⊗ Z (p) ∼ = Z (p) for all primes p, is the set of isomorphism classes of pseudo-integers. In other words,Ḡ(Z) is the set of isomorphism classes of torsion free abelian groups of rank 1 of ∞-free types [8, §42].
Proof. The spaces in the extended genus of S n which are (n − 1)-connected are characterized by the fact that the connecting homomorphism ∂ is surjective. In other words its kernel is a group of pseudo-integers. We conclude by Theorem 2.2.

A formula to compute localization genera
In the previous section we have been able to establish a complete and explicit list of all homotopy types in the extended rationalization genus of an odd sphere. In general, for arbitrary spaces and arbitrary localization functor, this is not to be expected. Following the approach of Dwyer, Kan, and Smith in [6] to classify towers of fibrations, we propose in this section a formula which we use later on to perform computations of "Postnikov" and "Neisendorfer" genus. We start with the necessary background from [6].
Let G be a space and consider the functor Φ which sends an object of Spaces ↓ B aut(G), i.e. a map t : X → B aut(G), to the twisted product X × t G. Dwyer, Kan, and Smith describe a right adjoint Ψ in [6,Section 4]. They find first a model for aut(G) which is a (simplicial) group and thus acts on the left on map(G, Z) for any space Z. This induces a map r : B aut(G) → B aut(map(G, Z)). The functor Ψ sends then Z to the projection map from the twisted product B aut(G) × r map(G, Z) → B aut(G). This allows right away to construct a classifying space for towers, in our case they will be of length 2.
where the homotopy fiber of p is G and that of q is H, is B aut(G) × r map(G, B aut(H)). The assumption that LLX is contractible is restrictive if we would require this for all spaces X.
This would amount to imposing that the localization functor L is a so-called nullification functor, i.e. a homotopical localization functor associated to a map of the form A → * such as a Postnikov section -when A is a sphere, [7, 1.A.6] -or Quillen's plus construction. We impose this condition however on a single space, and this happens sometimes for localization functors that are not nullifications. When X is a simply connected space of finite type and L is rationalization, p-completion,

Neisendorfer genus and Postnikov genus
Let P be the nullification functor [7, 1.A.4] with respect to the wedge BZ/p taken over all primes p. The unexpected effect of P on highly connected covers of finite spaces has been first studied by Neisendorfer, [18]. It relies on Miller's solution to the Sullivan conjecture, [15]. The name Neisendorfer's functor is commonly used for P BZ/p followed by completion at the prime p.

This composition of two localization functors however is not itself a localization functor because
it fails to be idempotent in general. Therefore we use here the name Neisendorfer's functor for localization with respect to the maps c : BZ/p → * and a wedge of universal mod p homology equivalences h. Thus L h coincides with profinite completion on nilpotent spaces andP X = L c∨h X often agrees with (P X) ∧ , e.g. when the latter space is already BZ/p-null.
As a direct consequence we compute the Neisendorfer genus of a finite complex. In a given localization genus set we fix the localization and the fiber of the localization map. Thus, even if all highly connected covers of a finite complex have the same Neisendorfer localization as the original complex, they do not belong to the (extended) genus set since the fibers of the localization maps fail to agree. Proof. Applying Theorem 4.1 to X with n = 1 we see that P X ≃ X.
For us the following consequences will be important. In particular point (3) will help compute the monoids of self-equivalences which appear in the formula from Corollary 3.4.
Corollary 4.3. Let X be a simply connected finite complex with π 2 (X) finite. Suppose that π >N (X) ⊗ Q = 0 for some integer N . Then [14] There are weak equivalences Proof. Since X N 0 is contractible we deduce from Theorem 4.1 that P (X N ) = X τ , which proves (1). Hence starting with X N we see that we get first X τ by applying P , then X ∧ by applying completion, and finally X N by taking the N -connected cover. As this last functor is a pointed one (it can be seen for example as cellularization with respect to the sphere S N +1 , [7, Example 2.D.6]), we obtain a chain of weak homotopy equivalences aut * (X N ) ≃ aut * (X τ ) ≃ aut * (X ∧ ) ≃ aut * (X ∧ N ). This shows (2).
Wilkerson's double coset formula for the genus set [21, Theorem 3.8] exhibits G(X) as double coset of the so-called π * -continuous self-equivalences of (X ∧ ) 0 under the left action of aut(X ∧ ) and the right action of aut(X 0 ). Clearly X and the Postnikov section X[N ] are rationally identical by assumption. Also the groups of self-equivalences are the same for X and its completion by part (2) of this corollary (which is for the pointed version but the spaces here are simply connected).
Thus all ingredients in Wilkerson's formula are identical for X and X[N ], which shows that the We note that G(X[N ]) can be computed from Zabrodsky's exact sequence when X 0 is an H-space [23,25], [13, Theorem 4].
We turn now to our first computation of Postnikov genus. If f : S N +1 → * is the constant map, , and L f is a functorial N -connected cover,L f X ≃ X N .    is obviously in the L-genus of S 3 . We will come back to this kind of example with a detailed computation in the next section. easy to see that the space CP 2 × S 3 also belongs to G [2] (S 2 × S 5 ). Of course the condition of the corollary are not fulfilled since neither π 2 S 2 , nor π 3 S 2 , are torsion.

It would be interesting to construct similar examples with higher Postnikov sections and at least
2-connected spaces, so the π 2 assumption is trivially fulfilled.

Self-equivalences of connected covers of a sphere
Our next goal will be to determine the n-th Postnikov genus of an odd sphere S n with n ≥ 3.
This will be done by using Theorem 3.3, which involves the computation of the space of selfequivalences of the n-th connected cover S n n . This section prepares the terrain for the genus computation in the next section and focuses on handy properties of aut(S n n ∧ p ). We write X ∧ p for the p-completion of X. Since S n n is a torsion space, it is weakly equivalent to the product of its p-completions S n n ∧ p . Now But since S n n ∧ p is p-complete and ( q =p S n n ∧ q ) ∧ p is contractible, we see that this mapping space is weakly equivalent to p map(S n n ∧ p , S n n ∧ p ). Therefore the subspace of self-equivalences also splits as a product aut(S n n ) ≃ p aut(S n n ∧ p ).
Because of the formulas in Theorem 3.3 we wish to understand certain mapping spaces into B aut(S n n ∧ p ) and start with a few elementary lemmas.
(2) we know that the spaces of pointed self-homotopy equivalences of the p-completed sphere and its n-connected cover coincide. By adjunction the pointed mapping space map * (ΣBZ/p, B aut * ((S n ) ∧ p )) is equivalent to map * (BZ/p, aut * ((S n ) ∧ p )). But aut * (S n n ∧ p ) consists of certain components of map * ((S n ) ∧ p , (S n ) ∧ p ) ≃ map * (S n , (S n ) ∧ p ) = Ω n (S n ) ∧ p , and all components of this iterated loop space have the same homotopy type, and they are BZ/p-local by Miller's Theorem [15].
The following is much easier to prove for the connected cover than for the p-completed sphere itself.
The homotopy groups of this inverse limit are all trivial since the towers we consider here consist of finite p-groups and multiplication by p (in particular all lim 1 terms vanish).
Even though we are not so sure whether B aut * (S n n ∧ p ) is p-complete, the previous lemmas allow us to understand maps out of K(Z, n).
In particular it is K(Z, n)-local.
Proposition 6.2. The space map * (K(Z, n), B aut(S n n ∧ p )) is homotopically discrete with Z ∧ p components.
Proof. Since there is a weak equivalence B aut * (S n n ∧ p ) ≃ B aut * ((S n ) ∧ p ) by Corollary 4.3.
(2), we have a fibration K(Z ∧ p , n) → B aut(S n n ∧ p ) → B aut((S n ) ∧ p ). The proposition follows now from the previous lemma.
We finally arrive at the unpointed mapping space.
It is in particular an infinite, countable set. Explicitely, G [n] (S n p ) is in bijection with the set N + of natural numbers with a disjoint base point * .
Proof. The homotopy fiber of the evaluation map(K(Z, n), B aut(S n n ∧ p )) → B aut(S n n ∧ p ) is homotopically discrete and identifies with Hom(Z, Z ∧ p ) by the last proposition. Moreover the fundamental group π 1 B aut(S n n ∧ p ) ∼ = π 0 aut(S n n ∧ p ) coincides with π 1 B aut * ((S n ) ∧ p ) ∼ = (Z ∧ p ) × , the p-adic units. Their action on the p-adic integers comes from the natural action on π n (S n ) ∧ p . Thus the components of the mapping space we are looking at is the quotient Z ∧ p /(Z ∧ p ) × . Let N = {0, 1, 2, . . .} be the set of natural numbers and N + the union of N with a disjoint base point * . The quotient Z ∧ p /Z × p is in bijection with the set N + because any non-zero p-adic integer can be uniquely written as p k u where k ∈ N and u is a unit [20]. The extended genus set has been identified as the quotient of this set under the action of ±1. However, since −1 in Z is sent in the p-adics to a unit, there are no further identifications.

Construction 6.4.
Here is an explicit way to construct the countable set G L (S n p ) of spaces Y with π n Y ∼ = Z and Y n ≃ S n n ∧ p . The fibration S n n ∧ p → S n p → K(Z, n) is classified by a map c : K(Z, n) → B aut(S n n ∧ p ). The proof of Theorem 6.3 shows that there is a bijection taking * to the constant map and the nonnegative integer k ∈ N to c • p k .
Define the space Y p, * to be S n n ∧ p × K(Z, n) and Y p,k , k ∈ N, to be the homotopy pull-back of K(Z, n) p k −→ K(Z, n) ← S n p , or, equivalently, the homotopy fibre of S n p → K(Z, n) → K(Z/p k , n), where the second map is reduction mod p k . The bijection is then given by We show now that one can detect which fake partially p-completed sphere one is considering by a simple cohomological computation. We will be more precise in the proof. Proof. By Construction 6.4 these fake partially completed spheres fit into a tower of fibrations with fibers K(Z/p, n − 1). Let ι k denote a generator of H n (Y p,k ; Z) ∼ = Z, chosen in such a way that the image of ι k under f * is pι k+1 .
At the prime 2 the algebra structure is sufficient to distinguish the fakes. Notice that the mod 2 reduction of ι 1 is a polynomial generator detected in the mod 2 cohomology of K(Z/2, n − 1), but ι 0 is an exterior generator. Therefore (2 k ι k ) 2 = 0, but (2 k−1 ι k ) 2 = 0. This shows that if Y is any (n − 1)-connected space with π n Y ∼ = Z and Y n ≃ (S n n ) ∧ 2 , and ι denotes a generator of H n (Y ; Z), then Y ≃ Y 2,k where k is the smallest integer such that (2 k ι) 2 = 0.
At an odd prime p, the mod p reduction of ι 1 is an exterior generator detected in the mod p cohomology of K(Z/2, n−1), but it has non-trivial integral Steenrod operations, such as P 1 β, acting on it. This operation is represented by classes of order p in H n+2p−1 (K(Z, n); Z) corresponding to the pair (P 1 ι n , βP 1 ι n ) in mod p cohomology. This shows that if Y is any space with π n Y ∼ = Z and Y n ≃ (S n n ) ∧ p , and ι denotes a generator of H n (Y ; Z), then Y ≃ Y p,k where k is the smallest integer such that P 1 (p k ι) = 0. Theorem 6.6. The extended Postnikov genus set G [n] (S n ) of homotopy types of spaces Y such that Y [n] ≃ K(Z, n) and Y n ≃ S n n is uncountable, in bijection with p N + , where the product is taken over all primes.
Proof. We apply Theorem 3.3 and the identification B aut(S n n ) ≃ B aut(S n n ∧ p ) obtained above. Theorem 6.3 shows that the set of unpointed homotopy classes [K(Z, n), B aut(S n n )] is uncountable, in bijection with p N + and it remains to identify the action of the finite group Aut(K(Z, n)) ∼ = Z/2. But we have seen that it is trivial at each prime since −1 is a unit in the p-adic integers. Hence the action is trivial.
Elaborating a little bit on Construction 6.4, one can explicitly construct all these fake spheres. where the spaces Y p,k have been constructed in Example 6.4 and the second arrow is given by the diagonal inclusion. The homotopy fiber of the map Y K → K(Z, n) is the product p S n n ∧ p ≃ S n n . The restriction to B aut(S n n ∧ p ) yields Y p,k which is classified by k p . This describes all spaces in G [n] (S n ).
Thus we have a good handle on all these fake spheres S n . What is so special about the good old S n among them? The answer is in Theorem 4.5.
Proposition 6.8. Let Y be a space such that Y [n] ≃ K(Z, n) and Y n ≃ S n n . Then, if Y is a finite complex, Y has the homotopy type of S n .
Finally we address the question of what happens when one changes the n-th Postnikov section for a higher one. The result will basically remain the same. An explicit computation would prove to be more difficult, but the concrete example of fake spheres we have produced serve equally well now. Proof. Since Y has been constructed so that Y n ≃ S n n ∧ p , the same is true for a higher connected cover. The claim about the m-th Postnikov section follows by choosing p > m − n + 3 2 so π * S n has no p-torsion in degrees < m.
This implies again that, for any m, there are uncountably many homotopy types of spaces which look like odd spheres through the eyes of the m-th Postnikov section and m-connected cover. We end the section with a related computation of the extended Postnikov genus of complex projective spaces.
Proof. Let C = CP n [2n + 1] be the (2n + 1)-st Postnikov section of CP n . There are fibrations K(Z, 2n + 1) → C → K(Z, 2) and S 2n+1 2n + 1 → CP n → C where S 2n+1 2n + 1 decomposes as p S 2n+1 2n + 1 ∧ p . Obstruction theory shows that [CP n , CP n ] → [C, C] is bijective so that self-maps of CP n and C are classified up to homotopy by their degrees in H 2 (−; Z). In particular, Aut(C) ∼ = Aut(CP n ) ∼ = Z × = {±1} has two elements. Let c : C → B aut(S 2n+1 2n + 1 ) be the classifying map for the standard CP n . We have seen that B aut(S 2n+1 2n + 1 ) splits as a product p B aut(S 2n+1 2n + 1 ∧ p ), and so the classifying map c decomposes as a product c = p c p • ∆, where ∆ : C → p C is the diagonal map and c p : C → B aut(S 2n+1 2n + 1 ∧ p ). For any sequence m = (m p ) p ∈ p Z of integers, let P n m be the space classified by c p • m p • ∆ : C → B aut(X p ). For example CP n and C × S 2n+1 2n + 1 correspond respectively to the constant sequences (1) and (0).
Consider now the restriction of one of the components c p •m p to the fiber K(Z, 2n+1). The degree m p map on CP n induces the degree m n p map on the cover S 2n+1 , so that this restriction corresponds to the class of m n p in the coset Z ∧ p /(Z ∧ p ) × we obtained in Theorem 6.3. In particular, for any choice m p = p k this restriction represents a different homotopy class in [K(Z, 2n + 1), B aut(S 2n+1 2n + 1 ∧ p )]. In fact for any choice k p ∈ N, the sequences m = (p kp ) yield an uncountable number of homotopy types of fake complex projective spaces. Indeed the spaces P n m are all distinct since the homotopy pull-back of P n m → C ← K(Z, 2n + 1) is homotopy equivalent to the fake sphere described by the sequence (nk p ) as in Construction 6.7.