ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES

FREDERICK R. COHEN* AND RAN LEVI

1. introduction

Consider the space X_{n} = ℝP^{∞}/ℝP^{n−1} together with the boundary map in the Barratt-Puppe sequence

X_{n} ^{}ΣℝP^{n−1}.

Francis Sergeraert and Vladimir Smirnov [6] have considered the low dimensional homotopy groups of X_{n} as well as their loop space homology. Their results are quite interesting, and their questions fit with several results that appeared previously in the literature [3, 8].

It is the purpose of this note to elaborate on a few of these remarks, as well as pointing out some natural associated questions and their connection to a recent result of Jie Wu [8], where he shows that the homotopy of the 3-sphere is a summand of the homotopy of ΣℝP^{2}. That result is a consequence of the structure considered here. Moreover, it will be shown here that the homotopy type of the spaces X_{n} is closely related to that of certain finite complexes described below.

Here are some concrete results, the first of which is an observation that was also pointed out by Broto [2].

Theorem 1.1. There is a fibration

S^{3}  ^{}X_{2}  ^{}K(ℤ, 2),

which is split after looping.

Notice that the calculation of loop space homology for X_{2} follows trivially from this splitting result and the known loop space homology for S^{3}.

Theorem 1.2. There is a 2-local fibration

A_{3}  ^{}X_{3}  ^{}BS^{3},

where A_{3} is the 6-skeleton of the Lie group G_{2}. Furthermore, there is a splitting

Ω_{0}^{4}X_{3} ≃ Ω^{3}S^{3} × Ω_{0}^{4}A_{3},

where Ω_{0}^{n} denotes the component of the constant map in an iterated loop space.

* Partially supported by the NSF.

The calculation of loop space homology for X_{3} is a bit more involved. A work of David Anick [1] gives the homology of ΩA_{3}. Let V denote the 5-skeleton of the Lie group G_{2}. Then A_{3} is obtained from V by attaching a single 6-cell. It is easy to see that H = H_{∗}(ΩV; 𝔽_{2}) (in fact with any coefficients) is a tensor algebra on two generators a and b of dimensions 2 and 4 respectively. The obvious inclusion induces a H-module structure on H_{∗}(ΩA_{3}; 𝔽_{2}). Let H⟨t⟩ denote the free associative H-algebra on one generator t in dimension 5. Thus H⟨t⟩ is isomorphic to a tensor algebra on a, b and t. Define a differential d on H⟨t⟩ by d(t) = a^{2} and d(a) = d(b) = 0. This turns H⟨t⟩ into a differential graded algebra and Anick’s theorem now gives

Theorem 1.3. The mod-2 loop space homology of A_{3} is isomorphic as an H-module to the homology of the differential graded algebra (H⟨t⟩, d) defined above. Moreover the Poincaré series for H_{∗}(ΩA_{3}, 𝔽_{2}) is given by the formula

P_{ΩA_{3}}(t) = \frac{1 − t^{4}}{1 − t^{2} − t^{4} − t^{7}} .

The fibration

ΩA_{3}  ^{}ΩX_{3}  ^{}S^{3}

is not split because the connecting map S^{3} ^{}A_{3} is essential. However, it is degree 2 on the bottom cell and hence trivial on mod-2 homology. Since the fibration is multiplicative, the mod-2 Serre spectral sequence collapses at the E^{2}-page and one obtains

Corollary 1.4. There is an isomorphism of H_{∗}(ΩA_{3}, 𝔽_{2})-modules

H_{∗}(ΩX_{3}; 𝔽_{2}) ≅ H_{∗}(ΩA_{3}, 𝔽_{2}) ⊗ E[x_{3}].

Remark 1.5. The low dimensional homotopy of A_{3} is quite easy to compute, as the natural map A_{3} ^{}K(Z, 3) induces a homology isomorphism through dimension 6, and the homotopy fibre is easy to handle through low dimensions, as demonstrated below.

Recall the following theorem of J. Wu.

Theorem 1.6 (Wu). Let γ : P^{3}(2) ^{}BSO(3) denote the map given by inclusion to the bottom skeleton. Let Y denote its homotopy fibre. Then

1. Y is homotopy equivalent to Σ(ℝP^{4}/ℝP^{1}) ∨ P^{6}(2) and

2. After looping 4 times and restricting to components of the constant map Ω^{4}γ has a right homotopy inverse.

The boundary map in the Barratt-Puppe sequence

X_{n} ^{}ΣℝP^{n−1}

together with Theorem 1.2 and naturality gives a new proof part 2 the theorem. This theorem gives a way of computing the homotopy of P^{3}(2) through a considerable range, as it is given by that of the 3-sphere and Y.

Theorem 1.7. There is a fibration

S^{7} ∨ P^{6}(2) ^{}X_{4} ^{}BS^{3},

which splits after looping.

Notice that the loop space homology of A_{4} is the tensor algebra on the reduced homology of S^{7} ∨ P^{6}(2). It also follows that there infinitely many elements of order 8 in the homotopy groups of X_{4} [3]. It seems reasonable to conjecture that there does not exist an elements of order 16.

The maps described above fit in a more systematic context. Namely, consider the natural maps g_{n} : ℝP^{n} ^{}SO(n), which induce a surjection in cohomology. Using these maps one gets the following

Theorem 1.8. There are maps

α_{n} : X_{n} ^{}BSpin(n),

such that the fibre F_{n} of α_{n} is a finite complex.

One might be tempted to conjecture that the spaces X_{n} have homotopy exponents. In addition, one might wonder how the splittings of the loop space of the suspension of X_{n} impinge on features of the degree 2 map on spheres.

Indeed, the maps

X_{n} ^{}ΣℝP^{n−1}

yield factorisations of several useful maps related to the degree 2 map. In particular when n is even, the degree 2 map on S^{n} factors through X_{n} and ΣℝP^{n−1}, and when n is odd the map

P^{n+1}(2) ^{}S^{n},

given by collapsing onto the top cell followed by η factors through X_{n} and ΣℝP^{n−1}. A discussion on how to obtain these factorisations and a speculation on their possible utility is in the last section of the paper.

The work described in this note started when the authors were both visiting the CRM during the emphasis semester in spring 1998. Both authors take the pleasure of expressing their thanks to the CRM for its kind and generous hospitality.

2. The spaces X_{n}

Consider the natural map from ℝP^{∞} to K(ℤ, 2), given by the first non-vanishing integral cohomology class. Since K(ℤ, 2) is simply-connected this map factors through X_{2}. Thus there is a fibration

F ^{}X_{2} ^{}K(ℤ, 2).

Pulling this fibration back once, one obtains a principal fibration with fibre S^{1} and base space X_{2}. Inspection of the integral cohomology Serre spectral sequence for this fibration gives that the cohomology of the fibre is isomorphic to the cohomology of the 3-sphere. Since F is obviously simply-connected, the first statement of Theorem 1.1 follows.

To see that the fibration in the theorem splits after looping, observe that the connecting map

ΩK(ℤ, 2) = S^{1} ^{}F = S^{3}

is null-homotopic for the obvious reason. Thus the projection from ΩX_{2} to S^{1} has a section and the fibration splits.

Corollary 2.1. The torsion in the homotopy of X_{2} has an exponent at any prime p.

Next analyse X_{3} and X_{4}. Consider the inclusion i of ℤ/2ℤ as the centre of the Lie group S^{3}. This homomorphism induces a map

Bi : ℝP^{∞} = Bℤ/2ℤ ^{}BS^{3}.

In cohomology this map takes the generator u_{4} ∈ H^{∗}(BS^{3}, 𝔽_{2}) to z^{4}, where z ∈ H^{∗}(Bℤ/2ℤ, 𝔽_{2}) is the generator. Since BS^{3} is 3-connected, Bi factors through both X_{3} and X_{4}. Thus there are fibrations for i = 3, 4

A_{i}  ^{}X_{i}  ^{}BS^{3}. (1)

The cohomology of A_{i} in both cases is easy to compute using the Serre spectral sequence for the pulled back fibrations

S^{3}  ^{}A_{i}  ^{}X_{i}. (2)

This can be done either with mod-2 coefficients or integrally. In either case the spaces A_{i} have only 2-torsion and their mod-2 cohomology is given by the proposition below. The explicit calculation is straight-forward and is left for the reader.

Proposition 2.2. The mod-2 cohomology of A_{3} is generated additively by classes a_{3}, b_{5} and a_{3}^{2}, with Sq^{2}a = b and Sq^{1}b = a^{2}.

The mod-2 cohomology of A_{4} is generated additively by classes x_{5}, y_{6} and z_{7}, with Sq^{1}x = y.

The analysis of A_{4} follows directly: It is clear by the cohomological calculation that its 6-skeleton is given by the Moore space P^{6}(2). The fibre of the obvious map P^{6}(2) ^{}K(ℤ/2ℤ, 5) gives the 5-connected cover of P^{6}(2) and makes it visible that π_{6}(P^{6}(2)) is isomorphic to ℤ/2ℤ and is generated by η on the bottom cell. Attaching a 7-cell to P^{6}(2) by this unique non-trivial element would have resulted in a Sq^{2} connecting the 5 and 7 cells in the mod-2 cohomology of cofibre. The cohomological structure of A_{4} thus implies the structure claimed in Theorem 1.7. That the fibration in the theorem splits after looping follows by pulling it back once and observing the the connecting map from S^{3} to the fibre A_{4} is null-homotopic for connectivity reasons.

Next analyse the homotopy type of A_{3}. The 5-skeleton of this space is seen to be given by Y = S^{3} ∪_{η} e^{5} by inspection of cohomology. The natural map Y ^{}K(ℤ, 3) induces an isomorphism on mod-2 cohomology up to dimension 5 and its fibre is the 4-connected cover of Y. By inspection of the associated Serre spectral sequence, π_{5}(Y) = ℤ and the cohomological structure implies that the attaching map is given by a multiple of the generator by an integer divisible by 2 exactly once. One can in fact verify by doing an integral Serre sequence calculation that the attaching map is exactly twice the generator and so the structure of A_{3} is thus determined. Notice that it has been shown here that the homotopy type of a simply-connected space with the mod-2 and integral cohomology of A_{3} is determined uniquely. Thus it follows that A_{3} is homotopy equivalent to the 6-skeleton of the Lie group G_{2} as claimed.

An old theorem of J. Harper [4] shows that the 2-local homotopy type of the Lie group G_{2} is determined by its mod-2 cohomology. The result here by contrast is obtained by very elementary methods and does not overlap with Harper’s calculation.

The splitting result claimed in Theorem 1.7 follows immediately from the fact that the connecting map S^{3} ^{}A_{4} = P^{6}(2) ∨ S^{7} is null-homotopic for connectivity reasons.

For the splitting result claimed in Theorem 1.2 one needs the following

Lemma 2.3. Let Z be a 2-connected space and let g : S^{3} ^{}Z be any map. If π_{4}(g) is trivial then Ω^{2}(2g) is null-homotopic when restricted to Ω^{2}S^{3}⟨3⟩ at the prime 2.

Proof. Let f denote 2g. Thus f is given by the composite

S^{3} ^{2}^{}S^{3} ^{g}^{}Z

Recall that the loops on the second Hilton-Hopf invariant Ωh_{2} : Ω^{2}S^{3} ^{}Ω^{2}S^{5} has order 2 in the group [Ω^{2}S^{3}, Ω^{2}S^{5}]. Thus the composition

Ω^{2}S^{3} ^{2}^{}Ω^{2}S^{3} ^{Ωh_{2}}^{}Ω^{2}S^{5}

is null-homotopic.

This remains true if all spaces are replaced by their 1-connected covers and so the composition

Ω^{2}(S^{3}⟨3⟩) ^{2}^{}Ω^{2}(S^{3}⟨3⟩) ^{Ωh_{2}}^{}Ω^{2}S^{5}

is null and 2 on Ω^{2}(S^{3}⟨3⟩) lifts to the fibre of Ωh_{2}, which is given by ΩS^{3}. Thus after passing to 1-connected covers Ω^{2}f is homotopic to a composition

Ω^{2}(S^{3}⟨3⟩) ^{l}^{}ΩS^{3} ^{Ωj}^{}Ω^{2}(S^{3}⟨3⟩) ^{Ω^{2}g}^{}Ω^{2}(Z⟨3⟩).

But notice that j takes the fundamental class in π_{3}(S^{3}) to η ∈ π_{3}(Ω(S^{3}⟨3⟩)), which under Ωg is taken, by hypothesis, to 0. Hence Ω^{2}gΩj is null-homotopic and so Ω^{2}f is null-homotopic, restricted to 1-connected covers.

Notice that in the fibration

S^{3} ^{δ}^{}A_{3} ^{}X_{3}

the fibre inclusion δ is degree 2 on the bottom cell. Also the map A_{3} ^{}K(ℤ, 3) corresponding to the bottom cohomology class induces an isomorphism on cohomology up to dimension 5. Hence π_{4}(A_{3}) = 0 and the conditions of Lemma 2.3 are satisfied. Thus looping twice and passing to universal covers, the map

Ω^{2}δ : Ω^{2}S^{3} ^{}Ω^{2}A_{3}

is null-homotopic, implying the splitting result of Theorem 1.2.

Finally analyse X_{n} for higher values of n. Consider a map g_{n} : ℝP^{n−1} ^{}SO(n), which induces the inclusion of generators in homology. Then there is an induced diagram where the left column is a cofibration and the right column is a fibration.

ℝP^{n−1}  \overline{}SO(n)

[figure: ℝP^{∞}; =; ℝP^{∞}]

[figure: X_{n}; α_{n}; BSpin(n)]

Consider the fibration

ℝP^{∞} ^{}BSpin(n) ^{}BSO(n).

Using either the well known calculation of the cohomology of BSpin(n) [5] or inspection of the Serre spectral sequence of this fibration, one observes that for some minimal positive integer r depending on n the class z^{2^{r}} ∈ H^{2^{r}}(RP^{∞})(𝔽_{2}) is an infinite cycle. Let u_{2^{r}} ∈ H^{2^{r}}(BSpin(n), 𝔽_{2}) be any class restricting to z^{2^{r}}.

Lemma 2.4. Let F_{n} denote the fibre of α_{n}. Then there is an isomorphism

H^{∗}(F_{n}, 𝔽_{2}) ≅ H^{∗}(X_{n}, 𝔽_{2})/(z^{2^{r}}) ⊗ T or_{H^{∗}(BSpin(n),𝔽_{2})/(u_{2^{r}})}(𝔽_{2}, 𝔽_{2})

of H^{∗}(X_{n}, 𝔽_{2})-modules. In particular F_{n} has the homotopy type of a finite complex.

Proof. Observe that α_{n}^{∗} takes any choice of u_{2^{r}} to the cohomology class in H^{∗}(X_{n}, 𝔽_{2}) inflating to z^{2^{r}} and that any class in H^{2^{r}}(BSpin(n), 𝔽_{2}) which is in the image of inflation from H^{∗}(BSO(n), 𝔽_{2}) is sent to 0 by α_{n}^{∗}. The calculation now becomes direct by using the Eilenberg-Moore spectral sequence for the fibration

F_{n}  ^{}X_{n}  ^{}BSpin(n)

and Smith’s ”big collapse theorem” [7].

3. Some applications and sample calculations

Proposition 3.1. For any positive integer n the degree 2 map on S^{2n} factors through X_{2n}. Also if n is odd then the map

P^{2n}(2) ^{}S^{2n−1},

given by collapsing onto the top cell followed by η factors through X_{2n+1}.

Proof. Let i : S^{2n} ^{}X_{2n} denote the inclusion to the bottom cell. Let δ : X_{2n} ^{}ΣℝP^{2n−1} denote the connecting map in the Barratt-Puppe sequence. Then by projecting to the top cell, one gets a map q : X_{2n} ^{}S^{2n}. It is immediate that δ is degree 2 in dimension 2n. Thus the first part of the proposition follows.

For any n the 2n-skeleton of X_{2n−1} is given by P^{2n}(2). By composing with δ and collapsing to the top cell, one gets a map P^{2n}(2) ^{}S^{2n−1}. Restricted to the bottom cell, this map is immediately seen to be degree 0 and is hence null-homotopic. Thus it factors through the top cell via a map α : S^{2n} ^{}S^{2n−1}. The map α can be either η or the null map.

Consider the commutative diagram of cofibrations

[figure: P^{2n}(2); X_{2n−1}; X_{2n+1}]

[figure: P^{2n}(2); =; ΣℝP^{2n−2}; C; f]

[figure: ∗; ΣℝP^{∞}; ΣℝP^{∞}; h]

Notice that C is (2n+1)-dimensional whereas X_{2n+1} is 2n-connected. Since ΣℝP^{2n−2} is (2n − 1)-dimensional, f induces the zero map on mod-2 cohomology. Thus h^{∗} surjects in all dimensions. Notice that additively, H^{∗}(C, 𝔽_{2}) is isomorphic to that of H^{∗}(ΣℝP^{2n}, 𝔽_{2}). Thus h^{∗} is in fact an isomorphism through dimension 2n + 1. Now, let σx^{2n−2} ∈ H^{2n−1}(ΣℝP^{∞}, 𝔽_{2}) denote the generator. Then

Sq^{2}(σx^{2n−2}) = σSq^{2}(x^{2n−2}) = σ(Sq^{1}(x^{n−1})^{2}).

Thus Sq^{2}(σx^{2n−2}) = 0 if n is even and is equal to σx^{2n} if n is odd. The claim now follows by commutativity of the diagram

[figure: P^{2n}(2); ΣℝP^{2n−2}; C; S^{2n}; α; S^{2n−1}; K]

using the fact that η is detected by Sq^{2}.

A remark related to the last proposition is the following. By suspending the factorisation of the degree 2 map on an even sphere one obtains a factorisation for the same map on an odd sphere.

S^{2n+1}  ^{}ΣX_{2n}  ^{}Σ^{2}RP^{2n−1}  ^{}S^{2n+1}.

Barratt’s distributivity formula for the degree 2 map on an odd sphere states that

Ω[2] = 2 + (Ω[ι_{2n+1}, ι_{2n+1}] ∘ H_{2}),

where the second summand is the second Hilton-Hopf invariant composed with the loops on the Whitehead square

ΩS^{2n+1} ^{H_{2}}^{}ΩS^{4n+1} ^{Ω[ι_{2n+1},ι_{2n+1}]}^{}ΩS^{2n+1}.

This identity holds in the non-abelian group [ΩS^{2n+1}, ΩS^{2n+1}]. One may wonder if this map is null-homotopic after looping 2n times more.

The splittings here do not seem to inform on the problem, but factoring the degree 2 map through the double suspension of the projective space might be useful with regards to it.

Next consider the space A_{3} obtained as the fibre of the map X_{3} ^{}BS^{3}, constructed above. The calculation of loop space homology becomes easy using Anick’s technique described in [1]. Specifically, A_{3} is obtained from V = S^{3} ∪_{η} e^{5} by attaching a 6-cell. The cohomology of A_{3} determines the attaching map f : S^{5} ^{}V. One observes easily that on mod-2 homology the adjoint ad(f) : S^{4} ^{}ΩV takes the generator to the element a^{2} ∈ H_{4}(ΩV, 𝔽_{2}). In this situation Anick’s theorem [1, 3.7] applies and the first claim of Theorem 1.3 follows. One also obtains the formula

\frac{1}{P_{ΩA_{3}}(t)} = \frac{1 + t}{P_{N}(t)} − \frac{t}{P_{ΩV} (t)} − t^{5},

where N is defined to be the quotient of the algebra H_{∗}(ΩV, 𝔽_{2}) by the two sided ideal generated by a^{2}. The calculation of the Poincaré series for ΩA_{3} thus follows from knowing the series for the algebra N.

Notice that there is a shost exact sequence of graded vector spaces

0 ^{}Σ^{4}T [a, b] ^{}T [a, b] ^{}N ^{}0.

From this exact sequence one obtains that

P_{N}(t) = P_{ΩV} (t) − t^{4}P_{ΩV} (t) = \frac{1 − t^{4}}{1 − t^{2} − t^{4}} .

Pluging this in the formula above and simplifying, the proof of Theorem 1.3 is complete.

Wu’s splitting theorem is also implied by the observations made here. We recall the relevant part of the theorem.

Theorem 3.2 (Wu). Let γ : P^{3}(2) ^{}BSO(3) denote the map given by inclusion to the bottom skeleton. Let Y denote its homotopy fibre. Then After looping four times and restricting to components of the constant map Ω_{0}^{4}γ has a right homotopy inverse.

Proof. Consider the following commutative diagram of fibrations:

[figure: A_{3}; X_{3}; BS^{3}; Y; P^{3}(2); γ; BSO(3)]

where the vertical map in the centre is the connecting map in the Barratt-Puppe sequence in the cofibration defining X_{3} and the right vertical map is the obvious one. Looping this diagram twice and taking connected components the right vertical map becomes an equivalence.

Thus looping four times and taking connected components there is a diagram

[figure: Ω_{0}^{4}A_{3}; Ω_{0}^{4}X_{3}; Ω_{0}^{3}S^{3}; Ω_{0}^{3}A_{3}]

[figure: Ω_{0}^{4}Y; Ω_{0}^{4}P^{3}(2); Ω_{0}^{4}γ; Ω_{0}^{3}S^{3}; =; Ω_{0}^{3}Y]

By Theorem 1.2 the top right horizontal map is null-homotopic. Hence the same applies to the bottom right horizontal map and the result follows.

Finally, the following low dimensional homotopy calculations are obvious from the splitting results presented here and well known facts on the homotopy groups of the spaces involved. These calculations are motivated by some computer oriented questions of Sergeraert. The program developed by Sergeraert and his coauthors however can certainly handle spaces much more general than the specific idiosyncratic cases here. The remarks here give a corroboration of those calculations.

Proposition 3.3. The 2-primary low dimensional homotopy of X_{3} is given as follows:

π_{i}(X_{3}) = \begin{cases}0 & i ≤ 2 \\ ℤ/2ℤ & i = 3 \\ 0 & i = 4 \\ ℤ/2ℤ & i = 5, 6 \\ ℤ/4ℤ ⊕ ℤ/2ℤ & i = 7 \end{cases}

The 2-primary low dimensional homotopy of x_{4} is given as follows:

π_{i}(X_{4}) = \begin{cases}0 & i ≤ 3 \\ ℤ & i = 4 \\ π_{i−1}(S^{3}) ⊕ ℤ/2ℤ & i = 5, 6 \\ π_{6}(S^{3}) ⊕ ℤ ⊕ ℤ/4ℤ & i = 7 \\ π_{i−1}(S^{3}) ⊕ (ℤ/2ℤ)^{3} & i = 8, 9 \\ π_{9}(S^{3}) ⊕ (ℤ/8ℤ)^{2} & i = 10 \end{cases}

Proof. That the space X_{3} is 2-connected and π_{3}(X_{3}) ≅ ℤ/2ℤ follows at once by looking at the cohomology of X_{3}. By Theorem 1.2, there is a splitting

Ω_{0}^{4}X_{3} ≃ Ω^{3}S^{3} × Ω_{0}^{4}A_{3},

where Ω_{0}^{n} denotes the zero component and A_{3} is the 6-skeleton of the Lie group G_{2}. Thus it suffices to compute the homotopy of A_{3} through the specified range. Notice that A_{3} is also the 6-skeleton of K(ℤ, 3). Hence the inclusion induces an isomorphism on integral homology though dimension 5 and so π_{i}(A_{3}) = 0 for i = 4, 5. The fibre of this inclusion is the 3-connected cover of A_{3}, which is seen by inspection of the Serre spectral sequence to be 6-connected with π_{7} given by ℤ/2ℤ. It follows that π_{4}(X_{3}) = 0 and π_{i}(X_{3}) are given by π_{i−1}(S^{3}) for i = 5, 6. Finally for i = 7 there is an exact sequence

π_{7}(A_{3}) ^{}π_{7}(X_{3}) ^{}π_{6}(S^{3}),

which is split because of Theorem 1.2. The result for X_{3} follows.

The calculation for X_{4} uses the splitting of Theorem 1.7

ΩX_{4} ≃ S^{3} × Ω(S^{7} ∨ P^{6}(2)).

Thus

π_{i}(X_{4}) ≅ π_{i−1}(S^{3}) ⊕ π_{i}(S^{7} ∨ P^{6}(2)) ≅ π_{i+1}(S^{3}) ⊕ π_{i}(S^{7}) ⊕ π_{i}(P^{6}(2))

through dimension 10. The right hand side equality follows from looking at the fibre of the inclusion S^{7} ∨ P^{6}(2) ^{}S^{7} × P^{6}(2), given by Σ(ΩS^{7} ∧ ΩP^{6}(2)) which is 10-connected. The result for X_{4} follows from the known homotopy of S^{7} and P^{6}(2) in this range.

The homotopy of the 3-sphere is of course known through a larger range than is dealt with here. With some extra effort one can work out a few more homotopy groups for A_{3} as well as the cross terms that occur in bigger dimensions arising from the homotopy groups of the wedge S^{7} ∨ P^{6}(2).

Corollary 3.4. There are infinitely many summands of ℤ/8ℤ in the homotopy of X_{4}.

This follows from a calculation by the J. Wu and the first author [3] for the homotopy of mod-2 Moore spaces. Explicit dimensions of an infinite family of ℤ/8ℤ summands are listed in this article.

Conjecture 3.5. There do not exist any elements of order 16 in the homotopy of X_{4}.

The conjecture fits with the Barratt conjecture for the Moore space. The 2-primary component of the homotopy of S^{3} has exponent 4. The best known exponent for the 2-primary homotopy of S^{7} is 32, but a conjecture of Barratt and Mahowald gives 8 as an upper bound.

References

[1] D. Anick, A counterexample to a conjecture of Serre, Ann. of Math. 115 (1982), 1-33

[2] C. Broto,

[3] F. R. Cohen and J. Wu, A remark on the homotopy groups of Σ^{n}ℝP^{2}, Cont. Math. 181 (1995), 65–81

[4] J. R. Harper, Unpublished

[5] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the Spinor groups, Math. Ann. 194, Springer-Verlag (1971), 197-212

[6] F. Sergeraert and V. Smirnov, To appear.

[7] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129, (1967), 58–93

[8] J. Wu, Thesis (1995), University of Rochester

Department of Mathematics, University of Rochester, Rochester, NY, 14627, U.S.A.

E-mail address: ran@maths.abdn.ac.uk

Department of Mathematical Sciences, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

E-mail address: cohf@math.rochester.edu