On Orbit Configuration Spaces and the Rational Cohomology of F (ℝP^{n}, k)

Miguel A. Xicoténcatl

Abstract. The purpose of this article is to study the orbit configuration space F_{ℤ_{2}}(S^{n}, k) of k-tuples of non-antipodal, distinct points in S^{n}. The integral cohomology of F_{ℤ_{2}}(S^{n}, k) is computed. In the rational case, the algebra of invariants under the natural action of (ℤ_{2})^{k}, is isomorphic to the cohomology of F (ℝP^{n}, k).

1. Introduction

Let M be a connected manifold and denote by F (M, k) the ordered configuration space of k distinct points in M, as defined in [5]. Configuration spaces have proved to be very important in algebraic topology and for this reason they have received a lot attention since their introduction by Fadell and Neuwirth. The cohomology of ordered configuration spaces is very well understood for M = ℝ^{n}, S^{n} and N × ℝ, see [3] and [4]. Also, a model for the rational homotopy type of F (M, k) was given recently in [6] and [8] when M is a smooth complex projective variety. However, not very much is known about the cohomology of F (M, k) for more general manifolds.

Analogues to configuration spaces for the case of manifolds with group actions have been also consider in [9]. Namely, if G acts freely on M, define the orbit configuration space by

F_{G}(M, k) = {(m_{1}, …, m_{k}) ∈ M^{k} | Gm_{i} ∩ Gm_{j} = ∅ if i ≠ j}.

where Gm denotes the orbit of m in M.

Let ℤ_{2} act on S^{n} via the antipodal map. In this article we study the orbit configuration space F_{ℤ_{2}}(S^{n}, k). In particular, we give a complete description of its cohomology algebra with coefficients in ℤ and ℚ, see Theorem 5.2. Similar results in this direction have been obtained independently by E. M. Feichtner. The space F_{ℤ_{2}}(S^{n}, k) admits also a natural action (ℤ_{2})^{k}-action and the quotient is homeomorphic to F (ℝP^{n}, k). In the rational case, it is also shown that

1991 Mathematics Subject Classification. 55R80, 55R05, 20J06.

H^{∗} (F (ℝP^{n}, k); ℚ) ≅ H^{∗} (F_{ℤ_{2}}(S^{n}, k); ℚ)^{(ℤ_{2})^{k}}

The article is organized as follows. In section 2 we recall some of the basic properties of orbit configuration spaces. Then we specialize to the case of M = S^{n} and G = ℤ_{2} and proceed to compute the cohomology of F_{ℤ_{2}}(S^{n}, k+1). This is done in several steps.

The fibre of the projection onto the first coordinate, π:F_{ℤ_{2}}(S^{n}, k + 1) ⟶ S^{n} is homeomorphic to the ℤ_{2}-orbit configuration space F_{ℤ_{2}}(ℝ^{n} − {0}, k), where the ℤ_{2} action on ℝ^{n} − {0} is given through the involution τ(x) = −x/|x|^{2}. In section 3 we calculate the cohomology algebra of F_{ℤ_{2}}(ℝ^{n}−{0}, k). The relations obtained here are analogues of the 3-term relation A_{s,u}A_{s,t} − A_{t,u}A_{s,t} + A_{t,u}A_{s,u} = 0 in the cohomology of F (ℝ^{n}, k) as described by F. Cohen [2], [3]. In addition to these, we obtain an extra relation.

Theorem 1.1. The integral cohomology of F_{ℤ_{2}}(ℝ^{n}− {0}, k) is generated, as a graded algebra, by the set (n − 1)-dimensional classes: A_{i,0}, i = 1, …, k and A_{i,j}, Ā_{i,j} for 1 ≤ j < i ≤ k, subject to the following relations:

(a) For i = 1, …, k and 1 ≤ t ≤ s,

A_{i,0}^{2} = A_{s,t}^{2} = Ā_{s,t}^{2} = 0.

(b) For 1 ≤ t ≤ s,

A_{s,0}A_{s,t} = A_{t,0}(A_{s,t} − A_{s,0})
A_{s,0}Ā_{s,t} = (−1)^{n}A_{t,0}(Ā_{s,t} − A_{s,0})
A_{s,t}Ā_{s,t} = (−1)^{n}A_{t,0}(Ā_{s,t} − A_{s,t})

(c) For 1 ≤ u < t < s ≤ k,

A_{s,u}A_{s,t} = A_{t,u}(A_{s,t} − A_{s,u})
A_{s,u}Ā_{s,t} = (−1)^{n}(A_{u,0} + A_{t,0} − A_{t,u})(Ā_{s,t} − A_{s,u})
A_{s,t}Ā_{s,u} = (−1)^{n}Ā_{t,u}(Ā_{s,u} − A_{s,t})
Ā_{s,u}Ā_{s,t} = A_{u,0}(Ā_{s,t} − Ā_{s,u}) + (−1)^{n}(A_{t,0} − A_{t,u})(Ā_{s,t} − Ā_{s,u})

In sections 4 and 5 we work out all the differentials in the Serre spectral sequence that converges to H^{∗}F_{ℤ_{2}}(S^{n}, k + 1). Finally, in section 6 the (ℤ_{2})^{k}- action is analyzed for some values of k. As a corollary, we get the rational cohomology of the corresponding F (ℝP^{n}, k).

2. Orbit configuration spaces

Let M be a manifold with a free action of a group G and let F_{G}(M, k) be the corresponding orbit configuration spaces as define in section 1. The main relation to ordinary configuration spaces is given by the following result which was proven in [9].

Proposition 2.1. Let G act on M such that the canonical projection M → (M/G) is a principal G-bundle. Then G^{k} acts on F_{G}(M, k) and F_{G}(M, k)/G^{k} ≈ F (M/G, k).

In particular, there is a principal G^{k}-bundle: G^{k} ⟶ F_{G}(M, k) ⟶ F (M/G, k) which can be used for homological calculations, as shown here and in section 6. More precisely, notice that the G^{k}-action on F_{G}(M, k) induces an action on cohomology, and we have

Theorem 2.2. Let G be a finite group acting freely on a manifold M and let 𝔽 be a field of characteristic 0 or prime to |G|. Then there is an isomorphism of algebras:

H^{∗} (F (M/G, k) ; 𝔽) ≅ H^{∗} (F_{G}(M, k) ; 𝔽)^{G^{k}}

where (−)^{G^{k}} denotes the module of invariants.

Proof. Since G^{k} acts properly on F_{G}(M, k), we can apply the spectral sequence for a covering (as given in [7] pg. 342 )

E_{2}^{p,q} = H^{p} (G^{k} ; H^{q}F_{G}(M, k))

which converges to H^{∗}F (M/G, k). The hypothesis on the characteristic of 𝔽, implies that multiplication by |G|^{k} is an isomorphism in 𝔽. Hence the cohomology of G^{k} (with coefficients in any 𝔽[G^{k}]-module), is concentrated in degree 0 and is isomorphic to submodule of invariants. The Theorem follows.

A useful result in the study of the homology of orbit configuration spaces is the following analogue of the Fadell and Neuwith fibrations for ordinary configuration spaces. For any natural number ℓ, let Q_{ℓ}^{G} ⊂ M be th union of ℓ distinct orbits. The next Lemma was proven in [9].

Lemma 2.3. Let M be a manifold with a properly discontinuous action of a group G and, such that the orbit space M/G is again a manifold. Then for ℓ ≤ k, the projection p:F_{G}(M, k) ⟶ F_{G}(M, ℓ) onto the first ℓ coordinates is a locally trivial bundle, with fibre F_{G}(M − Q_{ℓ}^{G}, k − ℓ).

For the rest of the paper, we specialize to the case M = S^{n} with the antipodal action of the group ℤ_{2}. In this case, F_{ℤ_{2}}(S^{n}, k) is the subspace of the k-fold product S^{n} × … × S^{n} which consists of all k-tuples (x_{1}, …, x_{k}) such that x_{i} ≠ ±x_{j}.

Identify S^{n} with ℝ^{n} ∪ {∞}, the one point compactification of ℝ^{n}, by using the stereographic projection and choose ∞ to be the base point. Let π: F_{ℤ_{2}}(S^{n}, k + 1) → S^{n} be the projection onto the first coordinate. By the Lemma above, π is a fibration with fibre 𝔽_{ℤ_{2}}(S^{n} − {±e_{n+1}}, k). A further analysis of the stereographic projection shows that the restriction of the ℤ_{2}- action to (S^{n} − {±e_{n+1}}) ≈ ℝ^{n}− {0} is given by the involution:

τ(x) = −\frac{x}{|x|^{2}} .

Therefore we get a fibration

F_{ℤ_{2}}(ℝ^{n}− {0}, k) ⟶ F_{ℤ_{2}}(S^{n}, k + 1) ⟶^{π} S^{n} (2.1)

whose Serre spectral sequence will be studied in sections 4 and 5. For this purpose, we use the rest of this and next section to calculate the cohomology of the fibre of (2.1).

For k = 1 we have F_{ℤ_{2}}(ℝ^{n} − {0}, 1) ≃ S^{n−1}. Assume inductively that H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k − 1) is concentrated in dimensions multiples of (n − 1). A second application of Lemma 2.3 yields a fibration

\xymatrix{F_{ℤ_{2}}(ℝ^{n}− \{0\}, k) \ar[d]_{π_{1}} & (ℝ^{n}− \{0\} − Q_{k−1}^{ℤ_{2}}) \ar[l] \\ F_{ℤ_{2}}(ℝ^{n}− \{0\}, k − 1)}

whose fibre has the homotopy type of a wedge (⋁_{2k−1} S^{n−1}). From the induction hypothesis, it follows that all differentials in the Serre spectral sequence are trivial, and the cohomology of the total space splits as the tensor product of the cohomology of the base and the cohomology of the fibre. Proceeding inductively we get:

H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k) ≅ ⨂_{i=1}^{k} H^{∗}( ⋁_{2i−1} S^{n−1}).

Explicit generators in dimension (n−1) are constructed next. For i = 1, …, k and 1 ≤ t < s ≤ k, consider the maps

p_{i}, p_{s,t}, p̄_{s,t}:F_{ℤ_{2}}(ℝ^{n}− {0}, k) ⟶ S^{n−1}

given by:

p_{i}(x) = \frac{x_{i}}{|x_{i}|}
p_{s,t}(x) = \frac{x_{s} − x_{t}}{|x_{s} − x_{t}|}
p̄_{s,t}(x) = \frac{x_{s} − τx_{t}}{|x_{s} − τx_{t}|}

and use them to define cohomology classes:

A_{i,0} = p_{i}^{∗}(ι_{n−1})
A_{s,t} = p_{s,t}^{∗}(ι_{n−1})
Ā_{s,t} = p̄_{s,t}^{∗}(ι_{n−1})

where ι_{n−1} is a fixed generator of H^{n−1}(S^{n−1}). Then one verifies that all these classes are linearly independent. One possibility is to construct maps S^{n−1} ⟶ F_{ℤ_{2}}(ℝ^{n} − {0}, k) representing their duals in homology. Thus the classes defined above form a basis for H^{n−1}F_{ℤ_{2}}(ℝ^{n} − {0}, k) and the next theorem follows.

Theorem 2.4. The cohomology of F_{ℤ_{2}}(ℝ^{n}− {0}, k) is given additively by:

H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k) ≅ M_{1} ⊗ M_{2} ⊗ ⋯ ⊗ M_{k}

where M_{i} is the free module generated by the following set of (n−1)-dimensional classes {A_{i,0}} ∪ {A_{i,j}, Ā_{i,j}}_{1≤j<i}.

3. The algebra structure in H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k)

In order to prove the relations in Theorem 1.1, we have to compute the products among the generators of each factor M_{i} in the tensor product above. Since all the A_{i,j}’s are spherical classes, we already know that A_{i,j}^{2} = 0. The calculation can be reduced to the case k = 3 as follows. Remember that there is an isomorphism of modules

F_{ℤ_{2}}(ℝ^{n}− {0}, 3) ≅ ⟨A_{1,0}⟩ ⊗ ⟨A_{2,0}, A_{2,1}, Ā_{2,1}⟩ ⊗ ⟨A_{3,0}, A_{3,1}, Ā_{3,1}, A_{3,2}, Ā_{3,2}⟩

and make the following change of notation. For |j| < i ≤ k let

A_{i,j} = \begin{cases}A_{i,j} & if j ≥ 0 \\ Ā_{i,−j} & if j < 0. \end{cases}

Notice that for 1 ≤ u < t < s ≤ k, the projection

π_{s,t,u}:F_{ℤ_{2}}(ℝ^{n}− {0}, k) ⟶ F_{ℤ_{2}}(ℝ^{n}− {0}, 3)

given by π_{s,t,u}(x_{1}, …, x_{k}) = (x_{u}, x_{t}, x_{s}) is a fibration and admits a section. Therefore H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, 3) injects into H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k). Furthermore, the induced map satisfies

(π_{s,t,u})^{∗}:H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, 3) ⟶ F_{ℤ_{2}}(ℝ^{n}− {0}, k)

A_{1,0} ⟼ A_{u,0}
A_{2,0} ⟼ A_{t,0}
A_{3,0} ⟼ A_{s,0}
A_{2,±1} ⟼ A_{t,±u}
A_{3,±1} ⟼ A_{s,±u}
A_{3,±2} ⟼ A_{s,±t}

Thus, the problem has been reduced to calculate the products among the generators of H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, 3). And this is approached as follows. Each of the desired products is given by a linear combination

A · B = ∑ C_{s,t}^{i,j} A_{i,j} ⊗ A_{s,t}

where the product A_{i,j} ⊗ A_{s,t} runs over the natural basis for H^{2(n−1)} and coefficients C_{i,j}^{s,t} are to be determined. We construct maps

F_{s,t}^{i,j}:S^{n−1} × S^{n−1} ⟶ F_{ℤ_{2}}(ℝ^{n}− {0}, 3)

such that the induced morphisms in degree 2n − 2 are precisely the duals of the classes A_{i,j} ⊗ A_{s,t}. Under this circumstances,

C_{s,t}^{i,j} = (F_{s,t}^{i,j})^{∗}(A · B) = (F_{s,t}^{i,j})^{∗}(A) · (F_{s,t}^{i,j})^{∗}(B)

and then explicit calculations can be carried out in order to obtain the coefficients. The required maps are constructed as follows. Let e denote the first canonical unit vector in ℝ^{n} and define

F_{s,t}^{i,j}(z, w) = (y_{1}, y_{2}, y_{3})

where y_{1}, y_{2} and y_{3} are given by

y_{1} = \begin{cases}\frac{z}{2} & if i = 1 \\ e & otherwise \end{cases}

and for ℓ = 2, 3

y_{ℓ} = \begin{cases}y_{j} + \frac{z}{2} & if ℓ = i, j > 0 \\ τ(y_{|j|}) + \frac{z}{2} & if ℓ = i, j < 0 \\ \frac{z}{2} & if ℓ = i, j = 0 \\ y_{t} + \frac{w}{4} & if ℓ = s, t > 0 \\ τ(y_{|t|}) + \frac{w}{4} & if ℓ = s, t < 0 \\ \frac{w}{4} & if ℓ = s, t = 0 \\ ℓ · e & if ℓ ≠ i, s . \end{cases}

For example, F_{2,0}^{1,0} = (\frac{z}{2}, \frac{w}{4}, 3e), F_{3,−1}^{2,1} = (e, e + \frac{z}{2}, −e + \frac{w}{4}), etc. Then one must study the effect of all these maps on degree n−1. This analysis provides all the coefficients for the k = 3. The general result follows from naturality.

4. The computation of the differentials

In this section we determine the non trivial differentials in the Serre spectral sequence for the fibration (2.1), whose E_{2}-term is given by:

E_{2}^{p,q} = H^{p}(S^{n}) ⊗ H^{q}F_{ℤ_{2}}(ℝ^{n}− {0}, k) (4.1)

Since the cohomology of the fibre is generated multiplicatively by the classes in degree n − 1, it is enough to determine

d_{n}^{0,n−1} :E_{n}^{0,n−1} ⟶ E_{n}^{n,0} = H^{n}(S^{n}).

Let’s denote by x the n-dimensional class in H^{∗}(S^{n}) which corresponds to the natural orientation of S^{n}. Then we have:

Proposition 4.1. For 0 ≤ j < i ≤ k, the n-th differential in the spectral sequence (4.1) is given by:

d_{n}(A_{i,0}) = d_{n}(A_{i,j}) = d_{n}(Ā_{i,j}) = \begin{cases}0 & if n is odd \\ 2x & if n is even . \end{cases}

Proof. For n odd, the fibration (2.1) admits a section. Therefore the differential in question vanishes, and the spectral sequence collapses.

For n even, we first reduce the calculation to the cases k = 1 and 2. Let π_{i}: F_{ℤ_{2}}(ℝ^{n} − {0}, k) → ℝ^{n} − {0} be the projection onto the i-th coordinate. The induced map in cohomology satisfies π_{i}^{∗}(A_{1,0}) = A_{i,0}. Notice that π_{i} fits into a map of fibrations

\xymatrix{F_{ℤ_{2}}(ℝ^{n}− \{0\}, k) \ar[d] \ar[r]^{π_{i}} & ℝ^{n}− \{0\} \ar[d] \\ F_{ℤ_{2}}(S^{n}, k + 1) \ar[d] \ar[r]^{1×π_{i}} & F_{ℤ_{2}}(S^{n}, 2) \ar[d] \\ S^{n} \ar[r]^{id} & S^{n}}

Now, a comparison of the corresponding spectral sequences gives d_{n}(A_{i,0}) = d_{n} π_{i}^{∗}(A_{1,0}) = d_{n}(A_{1,0}). Similarly for i > j, consider the projections π_{i,j} : F_{ℤ_{2}}(ℝ^{n} − {0}, k) → F_{ℤ_{2}}(ℝ^{n} − {0}, 2) with π_{i,j}(x_{1}, …, x_{k}) = (x_{j}, x_{i}). These maps satisfy π_{i,j}^{∗}(A_{2,1}) = A_{i,j} and π_{i,j}^{∗}(Ā_{2,1}) = Ā_{i,j}. Again by naturality we have:

d_{n}(A_{i,j}) = d_{n}(A_{2,1})
d_{n}(Ā_{i,j}) = d_{n}(Ā_{2,1})

Secondly, we show that d_{n}(A_{2,1}) = d_{n}(Ā_{2,1}). For x ∈ ℝ^{n}, define its “conjugate” by x̄ = (x_{1}, …, x_{n−1}, −x_{n}). Let R : S^{n} → S^{n} be the rotation given by the following element of SO(n + 1):

[R] = (\begin{matrix}I_{n−1} & 0 \\ 0 & −I_{2} \end{matrix})

and let a : S^{n} → S^{n} be the antipodal map. Consider the map (a, 1, a) : F_{ℤ_{2}}(S^{n}, 3) ⟶ F_{ℤ_{2}}(S^{n}, 3) and the homotopic map R^{3} ∘ (a, 1, a). The later induces a map of fibrations:

\xymatrix{F_{ℤ_{2}}(ℝ^{n}− \{0\}, 2) \ar[d] \ar[r]^{φ} & F_{ℤ_{2}}(ℝ^{n}− \{0\}, 2) \ar[d] \\ F_{ℤ_{2}}(S^{n}, 3) \ar[d] \ar[r]^{R^{3}∘(a,1,a)} & F_{ℤ_{2}}(S^{n}, 3) \ar[d] \\ S^{n} \ar[r]^{R∘a} & S^{n}} (4.2)

In both columns of (4.2), the fibre over ∞ = e_{n+1} ∈ S^{n} = ℝ^{n} ∪ {∞} is the space F_{ℤ_{2}}(ℝ^{n}− {0}, 2). The induced map on the fibres is given by

φ(z, w) = (\frac{z̄}{|z|^{2}} , −w̄)

and it has the following effect in cohomology:

φ^{∗}(A_{1,0}) = −A_{1,0}
φ^{∗}(A_{2,0}) = (−1)^{n+1}A_{2,0}
φ^{∗}(A_{2,1}) = (−1)^{n+1}Ā_{2,1}
φ^{∗}(Ā_{2,1}) = (−1)^{n+1}A_{2,1}

These results are obtained by a direct analysis of the Serre spectral sequence for the fibration

ℝ^{n}− {0, ±e_{1}} ⟶ F_{ℤ_{2}}(ℝ^{n}− {0}, 2) ⟶ ℝ^{n}− {0}

Notice that the map between the bases in (4.2) induces multiplication by (−1)^{n+1} in H^{n}(S^{n}). Thus we obtain the following relation in the spectral sequence for the columns of (4.2)

d_{n}(Ā_{2,1}) = (−1)^{n+1}d_{n} φ^{∗}(A_{2,1})
= (−1)^{n+1}(R ∘ a)^{∗} d_{n}(A_{2,1})
= d_{n}(A_{2,1})

Next, we show that for n even, d_{n}(A_{1,0}) = 2x. Let V_{2}(ℝ^{n+1}) denote the Stiefel manifold of 2-orthonormal frames in ℝ^{n+1}, and recall from [9] that the inclusion of V_{2}(ℝ^{n+1}) in F_{ℤ_{2}}(S^{n}, 2) induces an equivalence of bundles between the unit tangent bundle of S^{n} and fibration (2.1) for k = 1. The only differential in the spectral sequence for the first is given by multiplication with the Euler class of S^{n} which, in this case, is just multiplication by 2.

Finally, we show that for n even, d_{n}(A_{2,1}) = d_{n}(Ā_{2,1}) = 2x. Consider the inclusion of F_{ℤ_{2}}(S^{n}, 3) into the ordinary configuration space F (S^{n}, 3). This map induces a map of fibrations

\xymatrix{F_{ℤ_{2}}(ℝ^{n}− \{0\}, 2) \ar[d] \ar[r]^{j} & F (ℝ^{n}, 2) \ar[d] \\ F_{ℤ_{2}}(S^{n}, 3) \ar[d] \ar[r]^{i} & F (S^{n}, 3) \ar[d] \\ S^{n} \ar[r]^{id} & S^{n}} (4.3)

which gives the usual inclusion j in the fibres. Recall that F (ℝ^{n}, 2) ≃ S^{n−1} and denote the generator in dimension (n − 1) by y. It is also well know [4] that in the spectral sequence for the second column d_{n}(y) = 2x. The result will follow from naturality if we show that j^{∗}(y) = A_{2,1}. To see this, look at the induced map of fibrations

\xymatrix{ℝ^{n}− \{0, ±e_{1}\} \ar[d] \ar[r]^{j_{2}} & ℝ^{n} − \{e_{1}\} \ar[d] \\ F_{ℤ_{2}}(ℝ^{n}− \{0\}, 2) \ar[d] \ar[r]^{j} & F (ℝ^{n}, 2) \ar[d] \\ ℝ^{n} − \{0\} \ar[r]^{j_{1}} & ℝ^{n}}

The inclusion of the fibres satisfies j_{2}^{∗}(y) = A_{2,1} and the claim follows.

5. The integral cohomology of F_{ℤ_{2}}(S^{n}, k + 1)

Through all this section, R will denote a principal ideal domain. The most important examples will be R = ℤ and ℚ. We have already proved that for n even:

d_{n}(A_{i,0}) = d_{n}(A_{s,t}) = d_{n}(Ā_{s,t}) = 2x

for 1 ≤ i ≤ k and 1 ≤ t < s ≤ k. We show here that this information suffices to determine all the differentials in the spectral sequence. For j ≥ 0 let

C_{j} = H^{j(n−1)} (F_{ℤ_{2}}(ℝ^{n}− {0}, k); R)

Then C_{∗} is a graded algebra and it is multiplicatively generated by C_{1}. Recall that a basis for C_{1}, as a free R-module, is given by:

B = (A_{1,0}, A_{2,0}, A_{2,1}, Ā_{2,1}, …, A_{k,0}, A_{k,1}, …, Ā_{k,1}, …).

At this point it will be convenient to make a change of basis. Set:

B_{i,0} = A_{i,0} − A_{1,0}  for 2 ≤ i ≤ k

B_{s,t} = A_{s,t} − A_{1,0}  for 1 ≤ t < s ≤ k
B̄_{s,t} = Ā_{s,t} − A_{1,0}

and define 𝔸 and 𝕂 to be the free R-modules generated by A_{1,0} and by the set {B_{i,0}, B_{s,t}, B̄_{s,t} | 2 ≤ i ≤ k, 1 ≤ t < s ≤ k}, respectively. Then we have C_{1} = 𝔸 ⊕ 𝕂 and since A_{1,0}^{2} = 0, it can be shown by induction that C_{j} = 𝔸 · 𝕂^{j−1} ⊕ 𝕂^{j} for j ≥ 1. Here 𝕂^{j} is the R-module spanned by the products of length j of elements in 𝕂 and 𝕂^{0} = R. Also, notice that 𝕂 is the kernel of the homomorphism:

d_{n}:E_{n}^{0,n−1} ⟶ E_{n}^{n,0}

Moreover, we have:

Lemma 5.1. For j ≥ 1, let d^{j} denote the differential d_{n} : E_{n}^{0,j(n−1)} → E_{n}^{n,(j−1)(n−1)}. Then

(a) ker(d^{j}) = 𝕂^{j} for j ≥ 1.

(b) im(d^{j}) = 2x · 𝕂^{j−1} for j ≥ 1 and in particular,

(c) im(d^{j+1}) = (2x) · ker(d^{j}).

Proof. Recall that E_{n}^{0,j(n−1)} = C_{j} = 𝔸 · 𝕂^{j−1} ⊕ 𝕂^{j}. Since 𝕂 = ker(d^{1}), it is possible to show by induction that d_{n}(𝕂^{i}) = 0 ∀i ≥ 0. In the other hand, let A_{1,0} · k be a typical element of 𝔸 · 𝕂^{j−1}. Then d_{n}(A_{1,0} · k) = d_{n}(A_{1,0}) · k = 2x · k. This last expression vanishes only when k does and the Lemma follows.

Therefore:

E_{n+1}^{0,j(n−1)} = 𝕂^{j}  for j ≥ 0,

E_{n+1}^{n,j(n−1)} = xA_{1,0} · 𝕂^{j−1} ⊕ (x · 𝕂^{j})_{2}  for j ≥ 0

where (−)_{2} : Free R-Mod ⟶ R-Mod is the “mod 2 reduction” functor, which takes ⨁_{i} R to ⨁_{i}(R/2R). Here 𝕂^{−1} is understood to be 0 and thus we have:

Theorem 5.2. The cohomology of F_{ℤ_{2}}(S^{n}, k + 1) is given by

(a) For R = ℤ or ℚ and n odd, there is an isomorphism of algebras:

H^{∗}F_{ℤ_{2}}(S^{n}, k + 1) ≅ H^{∗}(S^{n}) ⊗ H^{∗}F_{ℤ_{2}}(ℝ^{n}− {0}, k)

(b) For R = ℤ and n even:

H^{q} (F_{ℤ_{2}}(S^{n}, k + 1); ℤ) = \begin{cases}𝕂^{j} & if q = j(n − 1), j ≥ 0 \\ xA_{1,0} · 𝕂^{j−2} ⊕ (x · 𝕂^{j−1})_{2} & if q = j(n − 1) + 1, j ≥ 1 \\ 0 & otherwise. \end{cases}

(c) For R = ℚ and n even:

H^{q} (F_{ℤ_{2}}(S^{n}, k + 1); ℚ) = \begin{cases}𝕂^{j} & if q = j(n − 1), j ≥ 0 \\ xA_{1,0} · 𝕂^{j−2} & if q = j(n − 1) + 1, j ≥ 2 \\ 0 & otherwise. \end{cases}

Remark 5.3. In the case when R = ℚ, a simple calculation shows that

∑_{j=0}^{k} dim_{ℚ}(𝕂^{j}) t^{j(n−1)} = (1 + 3t^{n−1})(1 + 5t^{n−1}) … [1 + (2k − 1)t^{n−1}]

and thus, for n even the Poincaré polynomial of F_{ℤ_{2}}(S^{n}, k + 1) is given by

∏_{ℓ=2}^{k} [1 + (2ℓ − 1)t^{n−1}] + t^{2n−1} ∏_{ℓ=2}^{k} [1 + (2ℓ − 1)t^{n−1}]

= (1 + t^{2n−1}) ∏_{ℓ=2}^{k} [1 + (2ℓ − 1)t^{n−1}]

6. The (ℤ_{2})^{k}-action on F_{ℤ_{2}}(S^{n}, k)

Recall that (ℤ_{2})^{k} = ⟨ϵ_{1}, …, ϵ_{k}⟩ acts coordinate-wise on F_{ℤ_{2}}(S^{n}, k), namely for every x = (x_{1}, …, x_{k}) ∈ F_{ℤ_{2}}(S^{n}, k), ϵ_{i}(x) = (x_{1}, …, −x_{i}, …, x_{k}). Therefore, the maps ϵ_{i} induce a natural (ℤ_{2})^{k}-action in cohomology, and in the case of rational coefficients it was proved in section 2 that

H^{∗} (F (ℝP^{n}, k); ℚ) ≅ H^{∗} (F_{ℤ_{2}}(S^{n}, k); ℚ)^{(ℤ_{2})^{k}}

In this section we study the action for some examples. Notice that for i = 2, …, k the following diagram commutes:

\xymatrix{F_{ℤ_{2}}(S^{n}, k) \ar[d]_{π_{1}} \ar[r]^{ϵ_{i}} & F_{ℤ_{2}}(S^{n}, k) \ar[d]^{π_{1}} \\ S^{n} \ar[r]^{id} & S^{n}}

and then, ϵ_{2}, …, ϵ_{k} induce maps of fibrations. Their action on H^{∗}F_{ℤ_{2}}(S^{n}, k) is computed by analyzing the actions induced on the cohomology of the base and of the fibre.

The map ϵ_{1} also induces a map of fibrations, being a:S^{n} → S^{n} the induced map on the bases. Since a is not a pointed map, ϵ_{1} does not preserve fibres. Therefore, we need a separate argument for the case i = 1. Let R ∈ SO(n+1) be a rotation in ℝ^{n+1} that interchanges the points 0, ∞ ∈ S^{n}. For instance take R to be as in section 4. The restriction of R to S^{n} is ℤ_{2}-equivariant and moreover, it ℤ_{2}-equivariantly isotopic to the identity map 1_{S^{n}}. Thus it induces a map:

R^{k}:F_{ℤ_{2}}(S^{n}, k) ⟶ F_{ℤ_{2}}(S^{n}, k)

which is homotopic to the identity and such that the following diagram commutes:

\xymatrix{F_{ℤ_{2}}(S^{n}, k) \ar[d]_{π_{1}} \ar[r]^{R^{k}∘ϵ_{1}} & F_{ℤ_{2}}(S^{n}, k) \ar[d]^{π_{1}} \\ S^{n} \ar[r]^{R∘a} & S^{n}}

where a:S^{n} → S^{n} is the antipodal map. Thus R^{k} ∘ ϵ_{1} is a map of fibrations, which is homotopic to ϵ_{1} and hence it has the same effect in cohomology. Moreover, notice that R^{k} ∘ ϵ_{1} preserves fibres. Therefore we can express the action of ϵ_{1} on the spectral sequence using the expression for E_{2}^{∗,∗} in terms of H^{∗}(S^{n}) and the cohomology of the fibre over ∞.

For k = 2, the action of (ℤ_{2})^{2} is summarized in Table 1.

Table 1 \begin{aligned}& ϵ_{1} & ϵ_{2} \\ x & (−1)^{n+1}x & x \\ A_{1,0} & −A_{1,0} & (−1)^{n}A_{1,0} \end{aligned}

For n odd, H^{∗}F_{ℤ_{2}}(S^{n}, 2) = Λ(x) ⊗ Λ(A_{1,0}) and the invariants appear only on degrees 0 and n. Thus:

H^{∗} (F (ℝP^{n}, 2); ℚ) ≅ Λ(x).

For n even and k = 2, the module 𝕂 defined in last section, vanishes. Thus H^{∗}F_{ℤ_{2}}(S^{n}, 2) is exterior on the class x · A_{1,0}, which Since this class which is invariant under the (ℤ_{2})^{2}-action. Therefore

H^{∗} (F (ℝP^{n}, 2); ℚ) ≅ Λ(y_{2n−1}).

Remark 6.1. It is interesting to notice that for n even, this method takes care of the problem of the local coefficient system in the Fadell-Neuwirth fibration

(ℝP^{n} − Q_{1}) ⟶ F (ℝP^{n}, 2) ⟶ ℝP^{n}

If the action of π_{1}(ℝP^{n}) = ℤ_{2} on the fibre was trivial, we would have

E_{2}^{∗,∗} ≅ H^{∗}(ℝP^{n}) ⊗ H^{∗}(ℝP^{n−1})
≅ Λ(x_{n−1})

Actually, it can be shown that the ℤ_{2}-action on H^{∗}(ℝP^{n−1}) is trivial in degree 0 and multiplication by −1 on degree (n − 1). Therefore, the E_{2} term is given by the cohomology of ℝP^{n} with local coefficients

E_{2}^{p,q} = H^{p} (ℝP^{n}; \underline{H^{q}(ℝP^{n−1})}) (6.1)

which can be computed as follows. Let ℚ[τ] denote the group ring of ℤ_{2} = ⟨τ⟩, with the convention that τ^{2} = 1. Recall that the standard complex for S^{n} with two cells in every dimension is given as

ℚ[τ] ⟵^{1−τ} ℚ[τ] ⟵^{1+τ} ℚ[τ] ⟵^{1−τ} ⋯ ⟵^{1+τ} ℚ[τ] ⟵ 0 ⟵ ⋯

Applying the functor Hom_{ℚ[τ]}(−, ℚ), where ℚ has the trivial action, yields the complex

ℚ ⟶^{×0} ℚ ⟶^{×2} ℚ ⟶^{×0} ⋯ ⟶^{×2} ℚ ⟶ 0 ⟶ ⋯

and we obtain

E_{2}^{p,0} ≅ \begin{cases}ℚ & p = 0 \\ 0 & otherwise . \end{cases}

In the other hand, applying Hom_{ℚ[τ]}(−, ℚ(−1)) we get

ℚ ⟶^{×2} ℚ ⟶^{×0} ℚ ⟶^{×2} ⋯ ⟶^{×0} ℚ → 0 → ⋯

and then

E_{2}^{p,n−1} ≅ \begin{cases}ℚ & p = n \\ 0 & otherwise . \end{cases}

Therefore, the spectral sequence (6.1) collapses at E_{2} and provide us with the same answer for H^{∗}F (ℝP^{n}, 2).

Table 2 \begin{aligned}& ϵ_{1} & ϵ_{2} & ϵ_{3} \\ x & (−1)^{n+1}x & x & x \\ A_{1,0} & −A_{1,0} & (−1)^{n}A_{1,0} & A_{1,0} \\ A_{2,0} & −(A_{2,0} + A_{2,1} + Ā_{2,1}) & A_{2,0} & (−1)^{n}(A_{2,0} + A_{2,1} + Ā_{2,1}) \\ A_{2,1} & Ā_{2,1} & Ā_{2,1} & (−1)^{n−1}Ā_{2,1} \\ Ā_{2,1} & A_{2,1} & A_{2,1} & (−1)^{n−1}A_{2,1} \end{aligned}

Table 3 \begin{matrix}dim & Basis & Invariants \\ 0 & 1 & 1 \\ n − 1 & A_{1,0}, A_{2,0}, A_{2,1}, Ā_{2,1} & A_{2,1} + Ā_{1,0} \\ n & x & x \\ 2n − 2 & A_{1,0}A_{2,0}, A_{1,0}A_{2,1}, A_{1,0}Ā_{2,1} & 0 \\ 2n − 1 & xA_{1,0}, xA_{2,0}, xA_{2,1}, xĀ_{2,1} & x(A_{2,1} + Ā_{2,1}) \\ 3n − 2 & xA_{1,0}A_{2,0}, xA_{1,0}A_{2,1}, xA_{1,0}Ā_{2,1} & 0 \end{matrix}

For k = 3, the (ℤ_{2})^{3}-action on the E_{2} term is given in table 2.

For n odd, H^{∗}F_{ℤ_{2}}(S^{n}, 3) ≅ ⟨x⟩ ⊗ ⟨A_{1,0}⟩ ⊗ ⟨A_{2,0}, A_{2,1}, Ā_{2,1}⟩ and a basis for the invariants is given in table 3. Thus we have

H^{∗}F (ℝP^{n}, 3) ≅ Λ(x) ⊗ Λ(A_{2,1} + Ā_{2,1}).

For n even, the (ℤ_{2})^{3}-action on E_{2} induces a well defined action on E_{n+1}, which has the following picture:

[figure: ℚ; ℚ^{3}; ℚ; ℚ^{3}; 2n − 2; n − 1; n; O]

Let B_{2,0}, B_{2,1} and B̄_{2,1} stand for the classes of (A_{2,0} − A_{1,0}), (A_{2,1} − A_{1,0}) and (Ā_{2,1} − A_{1,0}) respectively. Then we have

Table 4

Basis  Invariants
E_{n+1}^{0,0}  1  1
E_{n+1}^{0,n−1}  B_{1,0}, B_{2,1}, B̄_{2,1}  0
E_{n+1}^{n,n−1}  [xA_{1,0}]  [xA_{1,0}]
E_{n+1}^{n,2n−2} [xA_{1,0}A_{2,0}], [xA_{1,0}A_{2,1}], [xA_{1,0}Ā_{2,1}]  0

Thus for n even, H^{∗}F (ℝP^{n}, 3) is an exterior algebra on a class of dimension (2n − 1). The results obtained in this section are summarized table 5. More details and computations for the cases k > 3 will appear elsewhere.

Table 5

X  H^{∗}(X; ℚ)  Poincaré polynomial
ℝP^{2k}  ℚ  1
F (ℝP^{2k}, 2)  Λ(y_{4k−1})  1 + t^{4k−1}
F (ℝP^{2k}, 3)  Λ(y_{4k−1})  1 + t^{4k−1}
ℝP^{2k+1}  Λ(x_{2k+1})  1 + t^{2k+1}
F (ℝP^{2k+1}, 2)  Λ(x_{2k+1})  1 + t^{2k+1}
F (ℝP^{2k+1}, 3) Λ(x_{2k+1}) ⊗ Λ(y_{2k}) (1 + t^{2k+1})(1 + t^{2k})

Acknowledgments

During the preparation of this work, the author was a visitor at the Max-Planck-Institut für Mathematik. He would like to thank the institution for its hospitality, support and excellent working environment.

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