The homotopy fibre of the inclusion $F_n(M) \longrightarrow \prod_{1}^{n} M$ for $M$ either $\mathbb{S}^2$ or $\mathbb{R}P^2$ and orbit configuration spaces

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The homotopy fibre of the inclusion F_n(M) −→ ^∏_1ⁿM for M either S² or RP² and orbit configuration spaces

Daciberg Lima Gonçalves, John Guaschi

To cite this version:

Daciberg Lima Gonçalves, John Guaschi. The homotopy fibre of the inclusionF_n(M)−→^∏_1ⁿM forM eitherS² orRP² and orbit configuration spaces. 2017. �hal-01627001�

The homotopy fibre of the inclusion

F

( M ) ֒ −→ ^∏

M for M either S

_or R _P

_and orbit configuration spaces

DACIBERG LIMA GONÇALVES

Departamento de Matemática - IME- Universidade de São Paulo Rua do Matão, 1010, CEP 05508-090 - São Paulo - SP - Brazil.

e-mail: [email protected] JOHN GUASCHI

Normandie Univ., UNICAEN, CNRS,

Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14000 Caen, France.

e-mail: [email protected] 27^th September 2017

Abstract

Let n≥^{1, and letι}n: Fn(M)−→^∏1ⁿ M be the natural inclusion of the n^thconfiguration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the mapιn, its homotopy fibre In, and the induced homomorphisms(ιn)_#kon the k^thhomotopy groups of Fn(M)and∏₁ⁿ M for k ≥ ¹in the cases where M is the2-sphereS² or the real projective plane R_P²_{. If k} ≥ 2, we show that the homomorphism (ιn)_#k is injective and diagonal, with the exception of the case n =k = 2and M =S², where it is anti-diagonal.

We then show that Inhas the homotopy type of K(Rn−1, 1)×^Ω(∏ⁿ₁⁻¹S²), where Rn−1is the(n−¹)^thArtin pure braid group if M =S², and is the fundamental group Gn−1of the (n−¹)^thorbit configuration space of the open cylinderS²\ {ez0,−ez0}with respect to the action of the antipodal map ofS²_{if M} = R_P²_{, whereez0} ∈ S². This enables us to describe the long exact sequence in homotopy of the homotopy fibration In−→ ^Fn(M)−→^{ιn ∏n}1 M in geometric terms, and notably the boundary homomorphismπ_k+1(∏ⁿ₁ M)−→ ^πk(In). From this, if M=R_P²_{and n}≥2, we show that Ker((ιn)_#1)is isomorphic to the quotient of Gn−1by its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order2generated by the centre of Pn(R_P²)that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5].

2010 AMS Subject Classification: 20F36 (primary); 55P15 Classification of homotopy type; 55Q40 Homotopy groups of spheres; 55R80 Discriminantal varieties, configuration spaces; 55R05 Fiber spaces

1 Introduction

Let M be a connected surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface, and let ∏₁ⁿ M = M× · · · ×^Mdenote then-fold Cartesian product ofMwith itself. Then^thconfiguration spaceofMis defined by:

F_n(M) = (

(x1, . . . ,x_n) ∈

∏n 1

M

x_i 6=x_jfor all 1≤^i,j≤^n,i 6= j )

.

It is well known that the fundamental groupπ₁(F_n(M))of F_n(M)is isomorphic to the pure braid group P_n(M)ofMonnstrings [FaN,FoN], and ifMis the 2-disc thenP_n(M)is the Artin pure braid groupP_n. Letι_n: F_n(M) −→ ^∏n1 Mdenote the inclusion map, and fork ≥ ^{0, let}(ι_n)_#k: π_k(F_n(M))−→ ^πk ∏₁ⁿ M

be the induced homomorphism of the corresponding homotopy groups. If no confusion is possible, we shall often just write ιin place ofι_n, andι# or(ι_n)#ifk =1. The homomorphism ι#: P_n(M) −→^π1 ∏ⁿ₁ M was studied by Birman in 1969 [Bi], and if M is a compact surface without boundary different from the 2-sphereS²and the real projective planeR_P², Goldberg showed that Ker(ι#) is equal to the normal closure in P_n(M) of the image of the homomorphism j#: P_n −→ ^Pn(M) induced by the inclusion j: D²−→ ^M of a topological disc D² _in M [Go]. In [GG5], we extended this result to S² _and R_P², and in the case of R_P²_{, we} proved thatι#coincides with Abelianisation, so that Ker(ι#)is equal to the commutator subgroupΓ₂(P_n(R_P²))ofP_n(R_P²).

The aim of this paper is to study further the mapsι_n and the induced homomorph- isms (ι_n)_#k in more detail in the cases where M = S² _or R_P², and to determine the homotopy type of the homotopy fibre ofι_n. Following [A, pages 91 and 108], recall that if f: (X,x₀) −→(Y,y₀)is a map of (pointed) topological spaces and Idenotes the unit interval then themapping pathof f, defined by:

E_f =ⁿ(x,λ) ∈ ^X×^{YI f}(x) = λ(0)^o (1) has the same homotopy type asX, the mapr_f: E_f −→ ^X given byr_f(x,λ) = xbeing a homotopy equivalence, and the map p_f: E_f −→ ^{Y defined by p}f(x,λ) = λ(1) is a fibration whose fibre is thehomotopy fibre I_f of f defined by:

I_f =ⁿ(x,λ) ∈ ^X×^{YI f}(x) = λ(0)andλ(1) = y0

o. (2)

We will refer to the sequence of maps I_f −→ ^X −→^{f Y} as the homotopy fibration of f, where the map I_f −→ ^X is the composition of the inclusion I_f −→ ^Ef by r_f. More details may be found in [A,Ha,W]. In what follows, we denote the homotopy fibre of ι_n byI_n. We shall determine the homotopy type of I_n in the cases whereM is eitherS² orR_P². This leads to a better understanding of the long exact sequence in homotopy of the fibration p_ιn, as well as an alternative interpretation of P_n(M) in terms of exact sequences. Additional motivation for our study comes from the fact that the higher homotopy groups of F_n(M) are known to be isomorphic to those ofS² _orS³ _{(see The-} orem5), so such exact sequences involve these homotopy groups.

In the rest of this section, let M be S² _or R_P². This paper is organised as follows.

In Section 2, we show in Lemma 7 that for all n,k ≥ 2, the homomorphism (ι_n)_#k is injective, and in Propositions9and10, we prove that this homomorphism is diagonal, with the exception of the case M = S² _{and n} = k = 2, where it is anti-diagonal.

The aim of Section3 is to prove Theorem 1stated below that describes the homotopy type of I_n. In Section 3.1, we define the framework and much of the notation that will be used in the rest of the paper. In the case ofR_P², this description makes use of certain generalisations of configuration spaces, which we now define. Letez0 ∈ S²_{, let} C =S²\ {ez₀,−ez₀}denote the open cylinder, and letτ: S² −→S²denote the antipodal map defined byτ(x) =−^xfor allx ∈ S²as well as its restriction to C. Ifn≥ 1, then^th orbit configuration spaceofCwith respect to the grouph^τi, which is a subspace ofF_n(C), is defined by:

F_n^hτi(C) = (

(x₁, . . . ,x_n) ∈

n

∏

1

M

x_i ∈/^xj,τ(x_j) for all 1≤i,j ≤n,i 6= j )

. Orbit configuration spaces were defined and studied in [CX], but some examples already appeared in [Fa,FVB]. LetG_n =π₁ F_n^hτi(C)denote the fundamental group ofF_n^hτi(C). In Lemma16, we show thatF_n^hτi(C)is a space of typeK(G_n, 1), and thatG_nmay be writ- ten as an iterated semi-direct product of free groups similar to that of the Artin combing operation ofP_n. We then have the following description of the homotopy fibre ofI_ιn. THEOREM 1.Let n ≥ 2 and let M = S² _or R_P². Then the homotopy fibre I_n of the map ι_n: F_n(M) −→ ^∏n1 M has the homotopy type of:

(a) F_n−1(D²)×^{Ω ∏n}₁⁻¹ S²_{if M}=S², or equivalently of K(P_n−1, 1)×^{Ω ∏n}₁⁻¹ S²_{, where} Ω ∏ⁿ₁⁻¹ S²denotes the loop space of∏ⁿ₁⁻¹ S²_.

(b) F_n^h₋^τi₁(C)×^{Ω ∏n}₁⁻¹ S²_{if M} =R_P², or equivalently of K(G_n−1, 1)×^{Ω ∏n}₁⁻¹ S²_. Theorem 1will be proved in Section3.2. The basic idea of the proof is to ‘replace’

ι_n by a map that is null homotopic and whose homotopy fibre has the same homotopy type as that of ι_n. The fact that this map is null homotopic implies that its homotopy fibre may be written as a product that is the homotopy fibre I_c of a constant map (see Remarks 14(b)). Another important tool is the relation between the homotopy fibres of fibre spaces and certain subspaces (see the Appendix, and Lemma35in particular).

Taking the long exact sequence in homotopy of the homotopy fibration ofι_n and using the results of Section 2, we obtain the following exact sequences, where for k ≥ 1,

∂_n,k: π_k(^∏n₁ M) −→^πk−¹(I_n)denotes the boundary homomorphism on the level ofπ_k associated to the homotopy fibration I_n −→ ^Fn(M) −→^{ιn ∏n}₁ M.

COROLLARY 2.Let n≥2, and let M =S²_orR_P²_.

(a) Suppose that k≥3(resp. M =S²_{and n}=k=2). Then we have the following split short exact sequence of Abelian groups:

1−→ ^πk(F_n(M))⁽−→^{ιn)#k π}k

ⁿ

∏

1

M _∂n,k

−→^πk−¹

Ωⁿ−¹

∏

1

S²−→ ^{1, (3)}

where the homomorphism(ι_n)_#kis diagonal (resp. anti-diagonal). Up to isomorphism, this short exact sequence may also be written as:

1 −→^πk(M) −→

∏n 1

π_k(M) −→

n−¹

∏

1

π_k(M) −→ ^1.

(b) Suppose that k =2, and that n≥^{3if M} =S². Then we have the following exact sequence:

1−→ ^π2

ⁿ

∏

1

M _∂n,2

−→ ^Rn−¹×^π1

Ωⁿ⁻¹

∏

1

S²−→ ^Pn(M) ⁽−→^{ιn)#1 π}1

ⁿ

∏

1

M

−→^1, (4) where R_n−1 = P_n−1 if M = S² _{and Rn}

−¹ = G_n−1if M = R_P², which up to isomorphism, may also be written as:

1−→ Zⁿ −→ ^Pn−¹⊕Zⁿ⁻¹−→ ^Pn(S²)−→ ^{1 if M} =S²

1−→ Zⁿ −→ ^Gn−¹×Zⁿ⁻¹−→ ^Pn(R_P²)⁽−→^ιn)#1 Zⁿ₂ −→ ^{1 if M} =R_P²_. In the case M =S², the short exact sequence does not split.

The homomorphisms that appear in the exact sequences of Corollary2can be made explicit. In order to understand better the homotopy fibration associated toι_nand these exact sequences, it is helpful to study the properties of F_n^hτi(C) and G_n, as well as the boundary homomorphism ∂_n,k, the casek = 2 being the most complicated due to the appearance of G_n−1inπ1(I_n). In Section 4, we analyseG_n. In Proposition19, we give a presentation, from which we deduce in Proposition20that the centre ofG_n is infinite cyclic, generated by an elementΘ_n similar in nature to the full-twist braid ofP_n(M).

If M = S² _(resp.R_P²_{), let n0} = 3 (resp. n0 = 2). In Section 5, we determine com- pletely the boundary homomorphism ∂n,2, that we will denote simply by ∂_n. One of the principal difficulties here is to describe concrete homotopy equivalences between I_n and the product spaces I_c that appear in the statement of Theorem 1and which are homotopy fibres of constant maps. We first introduce geometric representatives of gen- erators ofπ₁(^Ω(M))in Section5.1, and we use them both to describe certain elements ofπ1(I_n0)in Lemma24, and also to fix a basisB = (^eλ_ex0,eλ_ex1, . . . ,eλ_exn−3,eλ_ez0,eλ_−ez0)(resp.

B = (λ_x0,λ_x1, . . . ,λ_xn−2,λ_z0)) of π₂(∏ⁿ₁ M) in equation (48). Here, xe₀,ex₁, . . . ,ex_n−2,ez₀ are certain basepoints of S² that are defined in Section 3.1, and x0,x1, . . . ,x_n−2,z0 are their projections in R_P² under the universal covering map from S² _to R_P²_{. In equa-} tion (49), for each elementeλ_y (resp. λ_y) of B, we define the elementδe_y (resp. δ_y) to be its image inπ₁(I_n)under∂_n. In Section5.2, in Lemma26, we construct explicit homo- topy equivalences between the intermediate homotopy fibres that were introduced in Section3.1whose composition yields a homotopy equivalence betweenI_c andI_n. Here we also require Corollary37that is proved in the Appendix. In Proposition 27, we in- troduce an elementτb_n ofπ₁(I_n)that under the projection fromπ₁(I_n)toP_n(M), is sent to the full-twist braid ∆²_n of P_n(M), and we relateτb_n² to the elements of the set ∂_n(B). In Section 5.3, we describe ∂_n in the case n = n0, the main results being Theorem 29 and Corollary 30. The proof of Theorem29 is geometric in nature, and makes use of Lemma 31 that is also used later on in the paper. The analysis of ∂_n when n > _{n0 is}

carried out in Section 5.4. The basic idea is to consider the possible projections of I_n onto I_n0. However, need to take care with the basepoints here, and we bring in to play various homeomorphisms ensure that they coincide with those of a standard copy of I_n0 as studied in Section5.3. The main result of Section5.4is the following.

THEOREM 3.Inπ₁(I_n), we have:

b τ_n² =

(∂_n(eλ_xe0 +· · ·+eλ_xen−3 +eλ_ez0 −^eλ_−ez₀) if M =S²

∂_n(λ_x0 +· · ·+λ_xn−2 +λ_z0) if M =R_P²_.

From this, we obtain Corollary33that describes∂_n completely, and that generalises Corollary30. This enables us to reprove several isomorphisms involvingP_n(S²), and to build upon the description ofΓ₂(P_n(R_P²))given in [GG5].

PROPOSITION 4.Let M =S²_orR_P², and let n≥ⁿ0. (a) If M =S²then there are isomorphisms:

Ker((ι_n)_#1) = P_n(S²) ∼=P_n−1/h^∆4_n−1i ∼=F_n

−²⋊(F_n

−³⋊(· · ·⋊(F₃⋊ F₂)· · ·))×Z_2. (5) (b) If M =R_P²then there are isomorphisms:

Ker((ι_n)_#1) =^Γ2(P_n(R_P²))∼=G_n−1/h^Θ2_n−1i ∼=F_2n−3⋊(F_2n−5⋊(· · ·⋊(F₅⋊ F₃)· · ·))×Z_2. (6) In each case, theZ₂-factor corresponds to the subgroup

∆²_n

of P_n(M). The result of part (a) is in agreement with [GG5, equation (2)].

Finally, in an Appendix, we prove Proposition36that relates the homotopy fibres of fibre spaces and certain subspaces, and that implies that one of the maps that appears in the betweenI_c andI_nis indeed a homotopy equivalence. This result seems to be well known to the experts, but we were not able to find a proof in the literature.

Some of the results and constructions of this paper have since been generalised in [GGG]. More precisely, if X is a topological manifold without boundary of dimen- sion at least three, under certain conditions, the homotopy type of the homotopy fibre of the inclusion mapι_n: F_n(X) −→^∏n1 X was determined, and was used to study the cases where either the universal covering of X is contractible, or X is an orbit space S^k_/Gof a tame, free action of a Lie groupGon thek-sphereS^k. A complete description of the long exact sequence in homotopy of the homotopy fibration ofι_n, similar to that of Corollary 2, was given in the case X = S^k/G, where the group G is finite and k is odd. The authors have also written a survey that summarises the current situation, and that includes some questions and open problems about graph and (orbit) configuration spaces [GG6].

Acknowledgements

The authors are grateful to Michael Crabb for his help with the results of the Ap- pendix, and especially for proposing proofs of Lemma 35 and Proposition 36. This work on this paper started in 2013 initially as part of the paper [GG5]. The authors were partially supported by the FAPESP Projeto Temático Topologia Algébrica, Geo- métrica 2012/24454-8 (Brazil), by the Réseau Franco-Brésilien en Mathématiques, by the CNRS/FAPESP project n^o 226555 (France) and n^o 2014/50131-7 (Brazil), and the CNRS/FAPESP PRC project n^o275209 (France) and n^o 2016/50354-1 (Brazil).

2 Properties of ιn and (ιn)_#k

In this section, we determine some properties ofι_n and of(ι_n)_#k, wherek ∈ N_{. IfXand} Yare topological spaces, and f,g: X−→ ^Yare maps betweenXandY, we write f ≃^g if f and gare homotopic, and we denote the homotopy class of f by[f].

We first state the following description of the homotopy type of the universal cov- ering of the configuration spaces ofS²_andR_P²_.

THEOREM 5 ([BCP, FZ], [GG3, Proposition 10(b)]).Let M = S² _orR_P², and let n ∈ N_. Then the universal covering ^

F_n(M)of F_n(M)has the homotopy type ofS²_{if M}=S²_{and n}≤2 or if M=R_P²_{and n} =1, and has the homotopy type of the3-sphereS³_otherwise.

REMARK 6. IfM =S²_{(resp. M} = R_P²), then by [FVB, Corollary, page 244] (resp. [VB, Corollary, page 82]),π₂(F_n(M))is trivial for alln ≥^{3 (resp. n}≥^2).

Let M = S² _or R_P² _{and let n} ≥ ^{2. For i} ∈ {^{1, . . . ,n}}^{, let p}i: F_n(M) −→ ^{M and} e

p_i: ∏₁ⁿ M−→ ^Mdenote the respective projections onto the i^thcoordinate. If 1 ≤ ⁱ <

j≤n, letα_i,j: F_n(M) −→ ^F2(M)andeα_i,j: ∏ⁿ₁ M −→^∏2₁ ^Mdenote the respective pro- jections onto the i^thand j^thcoordinates. Observe that the maps p_i, ep_i, α_i,j and eα_i,j are fibrations, and that:

p_i =p_ei◦^ιn andι2◦^αi,j =_eαi,j◦^ιn. (7) Let 1 ≤ ⁱ < _j ≤ n. As maps from F_n(M) toM, we have pe_i◦^ιn = p₁◦^αi,j and pe_j◦^ιn = e

p₂◦^ι2◦^αi,j, and using (7), we obtain the following commutative diagram:

F_n(M) F2(M) M

∏₁ⁿ M ∏²₁ M

M M.

p_j

α_i,j

ι_n

p_i p₁

ι2

p₂ e

p_i

eα_i,j

e p_j

e p₁

e p2

(8)

In all of what follows, letτ: S²−→ S²denote the antipodal map defined byτ(x) = −^x for allx∈ S²_{, and letπ:} S² −→R_P²denote the universal covering.

LEMMA 7.Let M =S²_orR_P², and let n≥^2.

(a) Let 1 ≤ ⁱ ≤ n. If either k ≥ 3, or k = n = 2 and M = S², the homomorphism (p_i)_#k: π_k(F_n(M)) −→ ^πk(M)is an isomorphism.

(b) Let k≥2. Then the homomorphism(ι_n)_#k: π_k(F_n(M)) −→^πk(∏ⁿ₁ M) is injective.

REMARK 8. Let n and k be as in Lemma 7(a). Theorem 5 implies that π_k(F_n(M)) ∼= π_k(M), and the lemma provides an explicit isomorphism.

Proof of Lemma7. Letn,k≥2.