A PRODUCT DECOMPOSITION OF Ω_{0}^{3}ΣRP^{2}

JIE WU

Abstract. We give a specific product decomposition of the base-point path connected component of the triple loop space of the suspension of the projective plane.

1. Introduction

In this paper, we give a specific product decompositions of the base-point path connected component of the triple loop space of the suspension of the projective plane, which is denoted by Ω_{0}^{3}ΣRP^{2}. Let RP_{a}^{b} = RP^{b}/RP^{a−1}, let X < n > be the n-connected covering of the space X and let P^{n}(2) = Σ^{n−2}RP^{2}, the n-dimensional mod 2 Moore space. In particular, P^{3}(2) = ΣRP^{2}. Our theorem is as follows.

Theorem 1.1. There is a homotopy equivalence

Ω_{0}^{3}P^{3}(2) ≃ Ω^{2}(S^{3}⟨3⟩) × Ω_{0}^{3}(P^{6}(2) ∨ ΣRP_{2}^{4})

localized at 2.

This theorem was first given in my thesis where we only gave a product decomposition of Ω_{0}^{4}(ΣRP^{2}). The author would like to thank his thesis advisor Fred Cohen for pointing out that our original decomposition for Ω_{0}^{4}(ΣRP^{2}) can be delooping. He showed that the power map

4 : Ω^{2}(S^{3}⟨3⟩) → Ω^{2}(S^{3}⟨3⟩)

is null localized at 2, his proof will be used in this paper. The application of this theorem to calculate the homotopy groups π_{r}(P^{3}(2)) for r ≤ 11 was given in my thesis [Wu]. An independent calculation for π_{∗}(P^{3}(2)) can be found out in [Mu]. The homotopy groups of ΣRP^{2} may be useful in low dimensional topology.

Research at MSRI is supported in part by NSF grant DMS-9022140.

2. Proof of the Theorem 1.1

In this section, all spaces are localized at 2 and H_{∗}(X) means the homology with coefficients in field F_{2}, the field with two elements. Let j : P^{3}(2) → BSO(3) be the inclusion of the 3-skeleton of BSO(3) and let X_{2} be the pull back in the diagram

BSO(2) → BSO(3)
↑  ↑
X_{2}  → P^{3}(2),

where the map BSO(2) → BSO(3) is induced by the canonical injection SO(2) → SO(3). Thus there is a fibration

S^{2} → X_{2} → P^{3}(2).

Let i : S^{2} → X_{2} be the inclusion of the fibre.

Lemma 2.1. The map Ω^{2}(i) : Ω^{2}(S^{2}⟨3⟩) → Ω^{2}(X_{2}⟨3⟩) is null.

Proof: Consider the commutative diagram

→ π_{3}(BSO(3)) →^{0} π_{2}(S^{2}) ↪ π_{2}(BSO(2)) → π_{2}(BSO(3)) → 0
↑ ∥ ↑ ↑
→ π_{3}(P^{3}(2)) →^{0} π_{2}(S^{2}) ↪^{i_{∗}} π_{2}(X_{2}) → π_{2}(P^{3}(2)) → 0

where the top and bottom rows are long exact sequence of homotopy groups. Thus π_{1}(X_{2}) = 0, π_{2}(X_{2}) ≅ π_{2}(BSO(2)) = π_{2}(CP^{∞}) ≅ Z and i_{∗} : π_{2}(S^{2}) → π_{2}(X_{2}) is of degree 2. Let ϕ : S^{2} → X_{2} be a representive of the generator in π_{2}(X_{2}). Then there is a homotopy commutative diagram

\begin{matrix}S^{2} & →^{i} & X_{2} \\ ∥ & & ↑ ϕ \\ S^{2} & →^{[2]} & S^{2}, \end{matrix}

where [2] : S^{2} → S^{2} is of degree 2. Now consider the homotopy commutative diagram

ΩS^{2}  →^{H_{2}} ΩS^{3}
Ω([2]) ↓  ↓ Ω([4])
ΩS^{2}  →^{H_{2}} ΩS^{3},

where H_{2} is the second James-Hopf map. Notice that S^{3} is an H-space. We have

Ω([4]) ≃ 4 : ΩS^{3} → ΩS^{3},

where 4 : ΩS^{3} → ΩS^{3} is the power map of degree 4. Consider the EHP sequence S^{3} → Ω(S^{3}⟨3⟩) →^{H_{2}} ΩS^{5}, the map S^{3} → Ω(S^{3}⟨3⟩) is a representative of the nontrivial element η in π_{4}(S^{3}) ≅ Z/2. Notice that the composite

S^{3} →^{η} Ω(S^{3}⟨3⟩) →^{2} Ω(S^{3}⟨3⟩)

is null. Thus the map 2 ≃ Ω(2) : Ω^{2}(S^{3}⟨3⟩) → Ω^{2}(S^{3}⟨3⟩) factors through Ω(H_{2}) : Ω^{2}(S^{3}⟨3⟩) → Ω^{2}S^{5} up to homotopy. By a result of Cohen [C1, or C2], the map

2 · Ω(H_{2}) : Ω^{2}(S^{3}⟨3⟩) → Ω^{2}S^{5}

is null. Thus the power map

4 : Ω^{2}(S^{3}⟨3⟩) → Ω^{2}(S^{3}⟨3⟩)

is null. Notice that the map Ω(H_{2}) : Ω^{2}(S^{2}⟨3⟩) → Ω^{2}(S^{3}⟨3⟩) is a homotopy equivalence. Thus Ω^{2}([2]) : Ω^{2}(S^{2}⟨3⟩) → Ω^{2}(S^{2}⟨3⟩) is null and so the boundary map Ω^{2}(i) : Ω^{2}(S^{2}⟨3⟩) → Ω^{2}(X_{2}⟨3⟩) is null, which is the assertion.

Corollary 2.2. There is a homotopy equivalence

Ω_{0}^{3}P^{3}(2) ≃ Ω^{2}(S^{3}⟨3⟩) × Ω_{0}^{3}(X_{2}).

Lemma 2.3. The (mod 2) homology of X_{2}⟨2⟩ is as follows:

H̄_{r}(X_{2}⟨2⟩) = \{ \begin{matrix}Z/2 & r=3,4,6 \\ Z/2 ⊕ Z/2 & r=5 \\ 0 & otherwise. \end{matrix}

Proof: Consider the homotopy commutative of fibre sequences

S^{2} → BSO(2) = CP^{∞} → BSO(3)
∥  ↑  ↑
S^{2} →  X_{2}  → P^{3}(2)
↑  ↑  ↑
∗ →  X_{2}⟨2⟩  →^{=} X_{2}⟨2⟩

Thus the homotopy fibre of X_{2}⟨2⟩ → P^{3}(2) is SO(3). The assertion follows by inspecting the Serre spectral sequence for the principle fibre sequence

SO(3) → X_{2}⟨2⟩ → P^{3}(2).

Recall thatH_{∗}(ΩP^{3}(2)) ≅ T (u, v) as algebras with dim(v) = 2 and Sq_{∗}^{1}v = u, where T (u, v) is the tensor algebra generated by u and v.

Lemma 2.4. H_{∗}(Ω(X_{2}⟨2⟩)) ≅ T (V) as subalgebras of H_{∗}(ΩP^{3}(2)), where V has a basis

{u^{2}, [u, v], v^{2} + u[u, v], [u^{2}, v], [v^{2}, u] + u[u^{2}, v]}

in T (u, v) ≅ H_{∗}(ΩP^{3}(2)) and [x, y] is the commutator of x and y.

Proof: Notice that Ωj_{∗} : H_{∗}(ΩP^{3}(2)) → H_{∗}(SO(3)) is an epimorphism and

π_{1}(ΩP^{3}(2)) ≅ π_{1}(SO(3)).

Thus the Serre spectral sequence for the fibre sequence

Ω(X_{2}⟨2⟩) →^{Ωi} ΩP^{3}(2) →^{Ωj} SO(3)

collapses and therefore there is a short exact sequence of Hopf algebras

1 → H_{∗}(Ω(X_{2}⟨2⟩)) → H_{∗}(ΩP^{3}(2)) → H_{∗}(SO(3)) → 1.

Thus H_{∗}(Ω(X_{2}⟨2⟩)) is the cotensor product of F_{2} and H_{∗}(ΩP^{3}(2)) over H_{∗}(SO(3)) ( see [MM] ).

We first show that V ⊆ H_{∗}(Ω(X_{2}⟨2⟩)). Let ψ be the comultiplication in the Hopf algebra H_{∗}(ΩP^{3}(2)) ≅ T (u, v). Then ψ is determined by the generators.

ψ(u) = u ⊗ 1 + 1 ⊗ u

and

ψ(v) = v ⊗ 1 + 1 ⊗ v + u ⊗ u.

The comultiplication ψ on the basis for V is as follows:

ψ(u^{2}) = u^{2} ⊗ 1 + 1 ⊗ u^{2},
ψ([u, v]) = [u, v] ⊗ 1 + 1 ⊗ [u, v],
ψ(v^{2} + u[u, v]) = (v^{2} + u[u, v]) ⊗ 1 + 1 ⊗ (v^{2} + u[u, v]) + u^{2} ⊗ u^{2},
ψ([u^{2}, v]) = [u^{2}, v] ⊗ 1 + 1 ⊗ [u^{2}, v],

and

ψ([u, v^{2}] + u[u^{2}, v]) = ([u, v^{2}] + u[u^{2}, v]) ⊗ 1 + 1 ⊗ ([u, v^{2}] + u[u^{2}, v]).

Notice that H_{∗}(SO(3)) ≅ E(u, v), the exterior algebra generated by u and v. Thus V ⊆ H_{∗}(Ω(X_{2}⟨2⟩)). Let ϕ : T (V) → H_{∗}(Ω(X_{2}⟨2⟩)) be the algebra map induced by the inclusion V ⊆ H_{∗}(Ω(X_{2}⟨2⟩)) and let A = Ωi_{∗} ∘ ϕ(T (V)). Then A is the subalgebra of H_{∗}(ΩP^{3}(2)) generated by V, which is also a tensor algebra. Notice that V → Q(A) is an isomorphism. Thus A ≅ T (V) ( see [CMN, Proposition 3.9]) and so ϕ : T (V) → H_{∗}(Ω(X_{2}⟨2⟩)) is a monomorphism. Notice that Poincare series

χ(H_{∗}(Ω(X_{2}⟨2⟩))) = χ(H_{∗}(ΩP^{3}(2)))/χ(H_{∗}(SO(3)))
= (1 − t − t^{2})^{−1}(1 + t + t^{2} + t^{3})^{−1} = (1 − t^{2} − t^{3} − 2t^{4} − t^{5})^{−1} = χ(T (V)).

Thus ϕ : T (V) → H_{∗}(Ω(X_{2}⟨2⟩)) is an isomorphism, which is the assertion.

Lemma 2.5. The homology suspension σ_{∗} : QH̄_{∗}(Ω(X_{2}⟨2⟩)) → H̄_{∗+1}(X_{2}⟨2⟩) is an isomorphism.

Proof: Let x_{2}, x_{3}, x_{4}, y_{4} and y_{5} be in H_{∗}(Ω(X_{2}⟨2⟩)) corresponding to u^{2}, [u, v], v^{2} + u[u, v], [u^{2}, v] and [v^{2}, u] + u[u^{2}, v] in T (u, v), respectively. Consider mod 2 Serre spectral sequence E_{∗,∗}^{r} for the principle fibre sequence

Ω(X_{2}⟨2⟩) → ∗ → X_{2}⟨2⟩.

Notice that, for ∗ ≤ 5, H̄_{∗}(Ω(X_{2}⟨2⟩)) has a basis

x_{2}, x_{3}, x_{2}^{2}, y_{4}, x_{4}, [x_{2}, x_{3}], x_{2}x_{3}, y_{5}.

Let a_{j+1} = σ_{∗}(x_{j}) and b_{j+1} = σ_{∗}(y_{j}). Then

d_{3}(a_{3}) = x_{2},
d_{3}(a_{3}x_{2}) = x_{2}^{2},
d_{3}(a_{3}x_{3}) = x_{2}x_{3},
d_{4}(a_{4}) = x_{3},
d_{4}(a_{4}x_{2}) = x_{3}x_{2},

and

E_{s,t}^{r} = 0

for r ≥ 6,s > 0,t > 0 and s + t ≤ 6. Thus there exists a_{5}, b_{5}, b_{6} ∈ E_{∗,0}^{2} such that d_{5}(a_{5}) = x_{4},d_{5}(b_{5}) = y_{4} and d_{6}(b_{6}) = y_{5}. The assertion follows.

Lemma 2.6. The element

x_{3} = [u, v]

in H_{3}(Ω(X_{2}⟨2⟩)) is spherical.

Proof: Consider the exact sequence of homotopy groups

π_{4}(P^{3}(2)) → π_{3}(SO(3)) → π_{3}(X_{2}⟨2⟩) → π_{3}(P^{3}(2)) → π_{2}(SO(3)) = 0.

Notice that π_{3}(SO(3)) = Z and π_{4}(P^{3}(2)) is torsion. Thus π_{3}(X_{2}⟨2⟩) = Z or Z⊕G for some 2-torsion abelian group G. Now X_{2}⟨2⟩ is 2-connected and F_{2} = H_{3}(X_{2}⟨2⟩; F_{2}) ≅ π_{3}(X_{2}⟨2⟩) ⊗ F_{2}. Thus π_{3}(X_{2}⟨2⟩) = Z and so π_{2}(Ω(X_{2}⟨2⟩)) = Z. Now the 3-skeleton (Ω(X_{2}⟨2⟩))_{(3)} of Ω(X_{2}⟨2⟩) is the homotopy cofibre of an attaching map ϕ : S^{2} → S^{2}. Notice that

H_{2}((Ω(X_{2}⟨2⟩))_{(3)}; Z) ≅ H_{2}(Ω(X_{2}⟨2⟩); Z) = Z.

Thus ϕ is of degree 0 and so the 3-skeleton (Ω(X_{2}⟨2⟩))_{(3)} ≃ S^{2} ∨ S^{3}. The assertion follows.

Proof of Theorem 1.1: By Corollary 2.2, it suffices to show that

X_{2}⟨2⟩ ≃ P^{6}(2) ∨ ΣRP_{2}^{4}.

We use the notation in Lemmas 2.4 and 2.5. By Lemma 2.4, Steenrod operations on H̄_{∗}(X_{2}⟨2⟩) are given by

Sq_{∗}^{2}a_{5} = a_{3}, Sq_{∗}^{1}a_{5} = a_{4}, Sq_{∗}^{1}b_{6} = b_{5}.

Thus

H̄_{∗}(X_{2}⟨2⟩) ≅ M ⊕ N

as modules over Steenrod algebra, where M is generated by a_{3}, a_{4}, a_{5} and N is generated by b_{5}, b_{6}. By Lemma 2.6, the element x_{3} is spherical in H_{3}(Ω(X_{2}⟨2⟩)). Let f : S^{3} → Ω(X_{2}⟨2⟩) so that f_{∗}(ι_{3}) = x_{3}, where ι_{3} is a generator for H_{3}(S^{3}). Consider the relative Samelson product for the multiplicative fibre sequence

Ω(X_{2}⟨2⟩) → ΩP^{3}(2) → SO(3)

( see [N1] ). There is a relative Samelson product

[f, E] : S^{3} ∧ RP^{2} = P^{5}(2) → Ω(X_{2}⟨2⟩),

where E : RP^{2} → ΩP^{3}(2) is the suspension. Notice that both ι_{3} and u are primitive. Then

[f, E]_{∗}(ι_{3} ⊗ u) = [f_{∗}(ι_{3}), u] = [[u, v], u] = [u^{2}, v]

in T (u, v). Let θ = σ ∘ Σ[f, E] be the composite P^{6}(2) → ΣΩ(X_{2}⟨2⟩) →^{σ} X_{2}⟨2⟩ and let u_{5} and v_{6} be the bases for H_{5}(P^{6}(2)) and H_{6}(P^{6}(2)), repectively. Then θ_{∗}(u_{5}) = σ_{∗}([u^{2}, v]) = b_{5} and Sq_{∗}^{1}θ_{∗}(v_{6}) = b_{5}. Thus θ_{∗}(v_{6}) = b_{6} which is the only nonzero element in H_{6}(X_{2}⟨2⟩) and so

θ_{∗} : H_{∗}(P^{6}(2)) → H_{∗}(X_{2}⟨2⟩)

is one to one. Let B^{5} be the homotopy cofibre of θ : P^{6}(2) → X_{2}⟨2⟩. Then H̄_{∗}(B^{5}) has a basis {a_{3}, a_{4}, a_{5}}. Notice that a_{4} = σ_{∗}(x_{3}) is spherical. Thus B^{5} is the homotopy cofibre of an attaching map ϕ : S^{4} → S^{3} ∨S^{4} = (X_{2}⟨2⟩)_{(4)}. Notice that the homotopy fibre of the pinch map X_{2}⟨2⟩ → B^{5} is 4-connected and π_{4}(X_{2}⟨2⟩) ≅ π_{4}(B^{5}). Thus the composite

S^{4} →^{ϕ} S^{4} ∨ S^{3} → X_{2}⟨2⟩

is null, or the injection S^{4} ∨ S^{3} → X_{2}⟨2⟩ extends to a map s : B^{5} → X_{2}⟨2⟩. Now the composite

P^{6}(2) ∨ B^{5} →^{θ∨s} X_{2}⟨2⟩ ∨ X_{2}⟨2⟩ →^{▽} X_{2}⟨2⟩

is a homotopy equivalence, where ▽ is the fold map.

Consider homotopy commutative diagram of cofibre sequences

S^{4} →^{ϕ} S^{3} ∨ S^{4} → B^{5} → S^{5}
∥ p ↓ ↓ ∥
S^{4} →^{[k]} S^{4} → P^{5}(k) → S^{5},

where p is the projection. Notice that Sq_{∗}^{1}a_{5} = a_{4}. Thus P^{5}(k) ≃ P^{5}(2) localized at 2 and k = 2k′ with k′ ≡ 1(2). Notice that Sq_{∗}^{2}a_{5} = a_{3}. Thus B^{5} is not homotopy equivalent to P^{5}(2) ∨ S^{3} and therefore the composite

S^{4} →^{ϕ} S^{3} ∨ S^{4} →^{p} S^{3}

is essential which is equal to η in π_{4}(S^{3}) = Z/2, where p is the projection. Thus the homotopy class [ϕ] = η + 2k′[ι] in π_{4}(S^{3} ∨ S^{4}) ≅ Z/2 ⊕ Z with generators η and [ι], where k′ ≡ 1(2). Notice that ΣRP_{2}^{4} is the homotopy cofibre of (η, [2]) : S^{4} → S^{3} ∨S^{4}. Thus B^{5} ≃ ΣRP_{2}^{4} and the assertion follows.

References

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[Mu] J. Mukai, Some homotopy groups of M^{3}, preprint.

[MM] J. Milnor and J. Moore, On the structure of Hopf Algebras, Ann. of Math. 81 (1965), 211-264.

[N1] J. Neisendorfer, Primary homotopy theory, Memoirs. AMS, 25 (1980).

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

MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA E-mail address: jwu@msri.org