Daciberg Lima GONC¸ALVES John GUASCHI

1 Introduction

Let A and Y be topological spaces,let f:A→Y be a continuous map,and let I denote the unit interval.If y0∈Y is a base point,the homotopy fibre Ifof f is defined by If={(a,λ)∈ A × YI??λ(0)=f(a)and λ(1)=y0}(see Section 4 for more information about If).The knowledge of Ifis a relevant ingredient for many applications,such as the possible factorisation of a map g:X →Y through A,leading to results in fixed point theory for suitable choices of A,Y and f(see[9]),as well as the study of the induced homomorphisms πi(A)→ πi(Y),where i≥ 0,that form part of a long exact sequence in homotopy involving πi(If).Intuitively,we are trying to measure how A differs from Y in some sense.At this level of generality,we cannot expect to obtain deep results,and we must restrict our attention to certain families of spaces and maps in order to make any meaningful progress.

In this paper,A will be a subset of Y,and f will be the inclusion map.Our aim is to study the case whereX is the n-fold Cartesian product of a topological space X,where n∈N.The subspacewill be one of the following:

(a)A is the n-th ordered con figuration space of X,defined by[6,10–11]:

We will often refer to Fn(X)as the “usual”con figuration space of X.If X is a surface,it is well known that π1(Fn(X))is isomorphic to the pure braid group Pn(X)of X on n strings(see[4,11–12]).If additionally X is the two-dimensional disc,then Pn(X)is the Artin pure braid group on n strings,denoted by Pn.

(b)A is the n-th ordered orbit con figuration space with respect to a free action of a group G on X,defined by[7,30]:

See Section 2 for more details.

(c)A is the n-th ordered graph con figuration space(X)associated to a graph Γ whose vertices are labelled by{1,···,n},that has no loops,and that possesses at most one edge between two vertices,where we define

This notion was defined in[2],using the notationSee Section 6.1 for more information about.

One may see thatIn what follows,if A is one of the above spaces,then we will letdenote inclusion,and if m ≥ 0,then(ιn)#m:will denote the induced homomorphism on the level of πm(relative to

The following classical result of Ganea illustrates nicely the type of answer that we would like to obtain to question(II).Let(X,x0)and(Y,y0)be pointed spaces,let X∨Y denote the wedge product of X and Y that we regard as a subspace of the Cartesian product X×Y,and let ΩX denote the loop space of X equipped with the base point x0.Ganea’s theorem describes the homotopy type of the homotopy fibre of the inclusion ι:X ∨Y → X ×Y.

Theorem 1.1(see[15,p.302],[16,Equation(6),p.448]and[26])IfXandYhave the homotopy type of a CW-complex,then the homotopy fibre of the inclusionι:X ∨Y → X ×Yhas the homotopy type ofΩX ∗ ΩY,where∗denotes the join operation.

One may define an unordered version of each of the three types of ordered con figuration spaces mentioned above.For the usual con figuration space Fn(X),the symmetric group Snon n letters acts freely on Fn(X)by permutation of coordinates,and the n-th unordered con figuration space of X is defined to be the quotient space Dn(X)=Fn(X)/Sn.In a similar manner,the n-th unordered orbit con figuration space of X is defined by(X)=(X)/Sn.Thirdly,for a graph Γ as given in(c)above,let H be a maximal subgroup of Snthat acts freely on(X)by permutation of coordinates.Then the n-th unordered graph con figuration space of X(with respect to H)may be defined to be the quotient space(X)=(X)/H.Each such subgroup H gives rise to an unordered graph con figuration space.Actually,the quotient space may be interesting even if H is not maximal,two examples being the usual con figuration spaces,and the quotients that give rise to“mixed”braid groups(see[20,22]).Some other information and remarks about the graph con figuration spaces and their relationship with the automorphisms of the graph may be found in Section 6.1.In contrast with the other types of con figuration space,the determination of the subgroups of Snthat act freely on(X)by permutation of coordinates,in particular the maximal subgroups,is not completely clear,and some work needs to be done to find them.Even for question(I),very little is known about the con figuration spaces.In the case where X is a surface,the usual unordered con figuration spaces have been widely studied(their fundamental groups are the full braid groups of X),but this is not the case so far for the unordered orbit and graph con figuration spaces.Question(II)remains largely open if X is arbitrary and A is an ordered con figuration space.Nevertheless,some progress has been made very recently in the case where A is the usual ordered con figuration space Fn(X),and X is either

(i)a surface without boundary,or

(ii)the orbit space of a sphere by a tame,free action of a compact Lie group G,or

(iii)a space that admits a contractible universal covering.some choice of base point).If m=1 then we will often just write ιn#for this homomorphism of fundamental groups if no confusion is possible.Two broad and important questions involving the pair of spaces(A,Y)are:

(I)describe the induced homomorphisms(ιn)#m,where m ≥ 0.An example of this may be found in[3,19],where it is proved that for surfaces other than S2and RP2,(ιn)#mis injective if m≥2,and if m=1,the image of the induced homomorphism is the normal closure of the image of Pnby the homomorphism induced by embedding a topological disc in the surface.

(II)determine the homotopy type of the homotopy fibre of the map ιn:

The aim of this paper is to give a survey of the state of the art of these types of questions,and to describe these recent results.It is worth mentioning that the ordered orbit con figuration spaces appear in a natural way in the study of the problems for the ordinary con figuration spaces.This provides extra motivation to study such spaces.The paper is comprised of five sections besides the introduction.In Section 2,we discuss the orbit con figuration spaces(X).In Proposition 2.1,we show that if G acts freely and properly discontinuously on a surface X different from the 2-sphere S2and the real projective plane RP2,then(X)is a K(π,1).In the rest of Section 2,we study the free action of Z2induced by the antipodal map τ:S2→ S2on the open cylinder C,which we interpret as the complement of the north and south poles in S2.In Propositions 2.2 and 2.3 respectively,we give a presentation of(C)and we describe the homotopy fibre of the inclusion mapIn Section 3,we describe the homomorphismwhere S is either S2or RP2.The case S=S2is much simpler,but its analysis aids us in the study of the case of S=RP2.If S=RP2,the kernel of ιn#may be written as the direct product of the cyclic subgroup of order 2 generated by the full twist braid of Pn(RP2)and a subgroup Lnthat may be described as an iterated semi-direct product of free groups(see Proposition 3.1 and Theorem 3.2).In Proposition 3.2,we determine the Abelianisation of Ln.This understanding of Ker(ιn#)enables us to prove the outstanding cases of a conjecture of Birman[3]in Theorem 3.1,which mirrors a similar well-known result for S2.In Section 4,we extend our analysis of the inclusion map,where S is either S2or RP2,and determine the homotopy type of the homotopy fibre of ιnin Theorem 4.2.If S=RP2,the description of the homotopy fibre makes use of the orbit con figuration spaces(C).This allows us to analyse the long exact sequence in homotopy associated to the fibration involving the homotopy fibre of ιn(see Corollary 4.1),including the boundary homomorphism,using an analysis of the homomorphisms induced by ιn(see Proposition 4.1).One interesting point about this long exact sequence is that the homotopy groups of S2occur not only as the homotopy groups of S,but also as those of Fn(S)since by Theorem 4.1,the universal covering of Fn(S)has the same homotopy type as S2or S3(depending on the value of n).In Section 5,we determine the homotopy type of the homotopy fibre of the inclusionwhere X satisfies one of the above conditions(ii)or(iii).The results are given in Propositions 5.1 and 5.2 respectively.In the latter case,if G is finite,we describe the long exact sequence in homotopy associated to the fibre sequence obtained by turning ιninto a fibration in Proposition 5.3,which extends the results of Corollary 4.1.These two results make use of a more general analysis of the homotopy fibre of ιngiven by Theorem 5.1 and Corollary 5.1.Finally,in Section 6,we make some remarks about the graph con figuration spaces defined in Equation(1.3),and we finish the paper with a list of open problems and questions.

2 Orbit Con figuration Spaces

The notion of orbit con figuration space was introduced in[30]and further studied in[7,31].It generalises the standard concept of con figuration space,and has been applied recently in[18,24]to understand certain homotopy fibres and long exact sequences(see Sections 4 and 5 respectively).In this section,we will discuss these spaces,as well their unordered version,and develop some basic properties that are closely related to our questions.

Let X be a topological space.Given a free action G×X →X of a group G on a topological space X,the n-th orbit con figuration spaceX)of X(with respect to G)is defined as in Equation(1.2).Note that if G is the trivial group then(X)coincides with Fn(X).The fundamental group π1((X))will be called the n-th orbit pure braid group of X(with respect to G).

Let 1 ≤ i1＜ ···＜ in−1≤ n be integers.If X is a manifold that admits a properly discontinuous action of a group G such that the orbit space X/G is again a manifold,then by[7,Lemma 4],the projectiononto the n−1 coordinates xi1,···,xin−1is a locally-trivial fibre bundle whose fibre may be identified with F1(XG({xi1,···,xin−1})).As in the case of the usual con figuration space,we may prove the following result.

Proposition 2.1IfXis a surface without boundary different fromS2andRP2,andGis a Lie group that acts freely and properly discontinuously onX,then the space(X)is aK(π,1).

Proof If G is discrete,the result follows from the fact that F1(XG({xi1,···,xin−1}))and X are K(π,1)’s.If G is not discrete,then it is a Lie group of dimension 1,and so contains S1as a subgroup.But this implies that either the surface X is the torus,in which case X is compact,or X is the open cylinder,i.e.,homeomorphic to S2×(0,1).This follows from the fact that the quotient is a one-dimensional connected manifold,so is either S1or the open interval.

Let x0∈ S2,let C denote the open cylinder S2{x0,−x0},and let τ:S2→ S2denote the antipodal map defined by τ(x)= −x,as well as its restriction to C.Then the cyclic group 〈τ〉of order 2 acts freely on S2and on C.As a special case of Proposition 2.1,if G=〈τ〉Z2then(C)is a K(π,1)(see also[24,Lemma 16]).A presentation of the fundamental group of(C),which we denote by Gn,was computed in[24,Proposition 19],and may be described as follows.We identify C with the open annulus(0,1)×S1in the complex plane,and if z=reiθ∈ C,we take its antipodal point z′to be(1 − r)ei(θ+π).For n ∈ N,we choose basepoints x1,···,xnthat lie on the negative part of the x-axis,and we letbe the corresponding antipodal points(see Figure 1).For j∈N,the free group π1(C{x1,···,xj−1},xj)of rank 2j− 1 admits a basis{ρj,i|0 ≤ i≤ 2j− 2},where for different values of i,the element ρj,iis represented by a pair of antipodal loops.In Figure 1,for clarity,we depict one loop of every representative pair for each generator ρj,i.In Figures 2–4,for each ρj,i,we indicate the corresponding pair of antipodal loops.Using these geometric representatives and Fadell-Neuwirth type fibrations involving the spaces(C),we obtain the following presentation of Gn.

Proposition 2.2Letn∈N.The following constitutes a presentation of the groupGn:

generating set:

relations:let1≤j＜k≤n.

Then

where

in the remaining cases;

(III)for allj≤i≤2j−2,

which is the casel=i−j+1,and

which is the casel=k+j−1.

Figure 1 Generators of π1(C{x1,···,xj−1,},xj).

Figure 2 The generator ρj,0.

Figure 3 The generator ρj,i,1 ≤ i＜ j.

Some other properties of Gnwill be highlighted in Section 4.As far as we know,this is the only case in which the fundamental group of an orbit con figuration space(that is not a standard con figuration space)has been computed.This allows us to understand better the homotopy fibre of the inclusion mapUsing the presentation of Gn

given by Proposition 2.2,we may prove the following result.

Proposition 2.3LetHndenote the kernel of the homomorphism

Then the homotopy fibre of the inclusionaK(Hn,1),and the groupHnis isomorphic to the normal closure inGnof{ρj,i}i0.

Proof Since πkfor all k1 and the homomorphismis surjective,the same is true for the homotopy groups of the homotopy fibre of.Therefore this homotopy fibre is a K(Hn,1).Note that the groupis isomorphic to Zn.From the definition of the generators ρj,i,i0(see Figures 3–4),it is clear that they belong toConversely,given an element wfor each 1≤j≤n,the exponent sum of ρj,0in w is zero.Therefore w may be written as a product of conjugates of the ρj,i,where i0,multiplied by a product of commutators of elements of{ρj,0}1≤j≤n.The result follows using the fact that the ρj,0commute pairwise by relation(I)of Proposition 2.2.

As we mentioned in Section 1,the symmetric group Snacts freely on(X),and we may define the n-th unordered orbit con figuration space of X(with respect to G)to be the quotient space(X)=(X)/Sn.So there is an n!-fold regular covering map(X)→(X).As in the case of the usual con figuration spaces,many properties of(X)may be deduced from those of(X).For example,if X is a surface different from S2and RP2,it follows from Proposition 2.1 that(X)is a K(π,1).We also define the n-th orbit braid group of X(with respect to G)to be the fundamental group π1((X)).

3 The Map ι and the Induced Homomorphism ι#for Surfaces

Let S 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 n∈N.In this section,we discuss recent results obtained in[23]regarding the inclusion map ιn:S and the induced homomorphismon the level of fundamental groups,and more particularly,the cases where S=S2or RP2.To simplify the notation,we will often just write ι and ι#if n is given.As we mentioned in Section 1, π1(Fn(S))Pn(S),the pure braid group of S on n strings(see[4,11,14]),from which one deduces that ιn#is surjective in a straightforward manner,and if S is the 2-disc D2then Pn(D2)is the Artin pure braid group Pn.If D2is a topological disc lying in the interior of S and that contains the basepoints of the braids then the inclusion jn:D2→S induces a group homomorphism jn#:Pn→Pn(S).This homomorphism is known to be injective if S is different from S2and RP2(see[3,19]).If β ∈ Bnthen we will denote its image j#(β)in Pn(S)simply by β.It is well known that the centre of Pnis in finite cyclic,generated by the full twist braid that we denote byIf we consideras an element of Pn(S2)or of Pn(RP2),it is the unique element of order 2 and generates the centre(see[17,20–21,23,27,29]).If G is a group then we denote its commutator subgroup by Γ2(G)and its Abelianisation by GAb,and if H is a subgroup of G then we denote its normal closure in G by 〈H〉G.If m ∈ N,Fmdenotes the free group of rank m.

The study of the homomorphismwas initiated by Birman in 1969(see[3]).In that paper,she said that she had conjectured that Ker(ιn#)= 〈Im(jn#)〉Pn(S)if S is a compact orientable surface,but then stated without proof that her conjecture is false if S is of genus greater than or equal to 1(see[3,p.45]).However,several years later,Goldberg proved that Birman’s conjecture is in fact true in both the orientable and non-orientable cases for compact surfaces without boundary different from S2and RP2(see[19,Theorem 1]).In connection with the study of Vassiliev invariants of surface braid groups,Gonz´alez-Meneses and Paris showed that Ker(ιn#)is also normal in the n-string full braid group Bn(S)of S,and that the resulting quotient is isomorphic to the semi-direct productwhere the action is given by permuting coordinates(their work was within the framework of compact,orientable surfaces without boundary,but their construction is valid for any surface S,see[25]).In the case of RP2,this result was reproved using geometric methods(see[28]).

If S=S2,Ker(ιn#)is clearly equal to Pn(S2),and so by[20,Theorem 4],it may be decomposed as

where the first factor of the direct product is torsion free,and the Z2-factor is generated byUsing Artin combing,one may show that Pn−3(S2{x1,x2,x3})may be decomposed as a repeated semi-direct product of the form Fn−2⋊(Fn−3⋊(···⋊F2)···).This may be compared with the usual Artin combing operation of Pn:

One of the aims of[23]is to resolve Birman’s conjecture for surfaces without boundary in the remaining cases,namely S=S2or RP2,and in the case S=RP2,to elucidate the structure of Ker(ιn#)and to obtain a decomposition analogous to that of Equation(3.1)for RP2.One may see easily that the isomorphism given in(3.1)may not be generalised directly to the case of RP2,since for all n≥2,Pn(RP2)possesses elements of order 4(see[27,29]),whilst the group Pn−m(S{x1,···,xm})is torsion free for all n ＞ m ≥ 1 and any surface S(including S2and RP2).

Proposition 3.1(see[23,Proposition 1])Letn∈N.

(a)(i)Up to isomorphism,the homomorphismιn#: π1(Fn(RP2))→ π1(RP2)coincides with Abelianisation.In particular,Ker(ιn#)= Γ2(Pn(RP2)).

(ii)Ifn ≥ 2,then there exists a torsion-free subgroupLnofKer(ιn#)such that

where the subgroupgenerated by the full twist is isomorphic toZ2.

(b)Ifn≥2,then any subgroup ofPn(RP2)that is normal inBn(RP2)and that properly containsKer(ιn#)possesses an element of order4.

Note that if n=1 then B1(RP2)=P1(RP2)Z2and Δ21is the trivial element,so parts(a)(ii)and(b)of Proposition 3.1 do not hold.Equation(3.3)is the analogue of Equation(3.1),and as we will indicate shortly,the subgroup Lnmay also be written as a repeated semi-direct product of free groups.

Proposition 3.1(a)(i)is proved by taking a generating set of Pn(RP2)of the form{Ai,j,τk|1≤ i＜ j≤ n and 1≤ k ≤ n},where{Ai,j}1≤i＜j≤nis a standard set of generators of Pn,and for each 1 ≤ k ≤ n,τkis represented by a braid that corresponds to a generator of π1(RP2)based at the k-th base point of Pn(RP2).It is easy to see geometrically that ιn#(Ai,j)is trivial for all 1 ≤ i＜ j ≤ n and that ιn#(τk)is sent to the corresponding generator of π1(RP2).Using a presentation of Pn(RP2)(see[21,Theorem 4]),one sees that the Abelianisation homomorphism has the same effect on these generators,which proves part(a).If n ≥ 3,Proposition 3.1(a)(ii)may be obtained by considering the Fadell-Neu wirth pure braid group short exact sequence(see[11,14]):

where p2#:Pn(RP2)→ P2(RP2)is the homomorphism given geometrically by forgetting all but the first two strings,and taking the restriction of p2#to Ker(ιn#).Using part(i)and the fact that P2(RP2)is isomorphic to the quaternion group of order 8(see[12]),it is straightforward to check that p2#(Ker(ιn#))=??which is isomorphic to Z2.One may also check that the kernel of the restriction of p2#to Ker(ιn#)admits a section given by sendingto the central elementand is equal to Ker(ιn#)∩ Pn−2(RP2{x1,x2}),which as we mentioned above,is torsion free.This intersection may be taken to be the group Lnin the statement.One may show that there are precisely 2n(n−2)subgroups that satisfy the conclusions of part(a)(ii)(see[23,Remark 14]).If S=S2or RP2,the decompositions(3.1)and(3.3)are used in the proof of[23,Theorem 5]to compute the virtual cohomological dimension of Bn(S),Pn(S)and the corresponding mapping class groups of S with n marked points.

Using Proposition 3.1 in the case of RP2,we may then prove Birman’s conjecture for S2and RP2.

Theorem 3.1(see[23,Theorem 2])LetS=S2orRP2,and letn≥1.Then

The proof of Theorem 3.1 is obtained by using the fact that{Ai,j}1≤i＜j≤nis a standard set of generators of Pn.The case S=S2is straightforward since S2is simply connected.In the case S=RP2,using this set of generators,one sees easily that 〈Im(jn#)〉Pn(S)⊆ Ker(ιn#).From the presentation of Pn(RP2)given by[21,Theorem 4],one may determine a set of normal generators of Γ2(Pn(RP2)),and then use Proposition 3.1(a)(i)to obtain the converse inclusion.

More detailed information may be obtained about the subgroup Ln=Ker(ιn#)∩Pn−2(RP2{x1,x2}).Using Fadell-Neuwirth short exact sequences,the group Pm(S{x1,···,xq})may be decomposed as a repeated semi-direct product of the form

where k is the rank of Pm(S{x1,···,xq})= π1(S{x1,···,xq}).Notice that this rank is also equal to q−1+(2−χ(S)),where χ(S)is the Euler characteristic of S.In particular,in(3.1),Pn−3(S2{x1,x2,x3})Fn−2⋊ (Fn−3⋊(···⋊F2)···).As we see in the following theorem,a similar result may be obtained for Ln.

Theorem 3.2(see[23,Theorem 3])Letn≥3,and consider the Fadell-Neuwirth short exact sequence(3.4).ThenLnmay be identified with the kernel of the composition

where the first homomorphism is that appearing inEquation(3.4).The image of this composition is the product of the lastn − 2copies ofZ2.In particular,Lnis of index2n−2inPn−2(RP2{x1,x2}).Further,Lnis isomorphic to an iterated semi-direct product of free groups of the formF2n−3⋊ (F2n−5⋊ (···⋊ (F5⋊ F3)···)).

The first part of Theorem 3.2 may be obtained by analysing the commutative diagram involving the short exact sequence(3.4)and its restriction to Ker(ιn#).In the repeated semidirect product decomposition of Ln,every factor acts on each of the preceding factors,and not just on the group formed by these factors.In this sense,the intersection of Ker(ιn#)with the repeated semi-direct product structure Pn−2(RP2{x1,x2})restricts to each of the factors in a nice way.This is also the case for Pn−2(RP2{x1,x2}),and implies an Artin combing-type result for these groups.Taking into account the actions,these repeated semi-direct products may be used to prove the following result about the Abelianisation of Pn−2(RP2{x1,x2})and Ln.

Proposition 3.2(see[23,Proposition 4])Ifn≥3,then

(a)(Pn−2(RP2{x1,x2}))AbZ2(n−2).

(b)(Ln)AbZn(n−2).

As we will see at the end of Section 4,Lnis closely related to the fundamental group of an orbit con figuration space of the open cylinder.

4 Recent Results about the Homotopy Fibre and the Long Exact Sequence where A is a Con figuration Space and X is a Surface

We start this section by recalling the construction of the homotopy fibre of a continuous map f:A→Y of topological spaces that allows us to turn f into a fibration.Let y0be a base point of Y,and let

denote the mapping path and homotopy fibre of f respectively.It is well known that If→Ef→ Y is a fibration,and that Efand A have the same homotopy type,a homotopy equivalence between Efand A being given by the projection Ef→ A onto the first coordinate(see[1,Proposition 3.5.8 and Remark 3.5.9]).Let pf:If→ A denote the composition of the inclusion If→ Efby this projection.We will refer to the sequence of mapsas a homotopy fibration.By[1,Proposition 3.3.17],the homotopy fibres of two homotopic maps have the same homotopy type,and if f is null homotopic then Ifhas the same homotopy type as A×ΩY,where ΩY denotes the loop space of Y.These two properties will be useful in what follows.Notice that Ifis determined by the homotopy pull-back diagram

where cy0:∗→Y is the constant map determined by a point y0∈Y,and q:If→∗is the constant map(see[1,p.205]).

The main aim of[24]is to determine the homotopy type of the homotopy fibre of f in the case where S=S2or RP2,A=Fn(S),and f is the inclusion map ιn:Fn(S)As a consequence,this leads to a better understanding of the induced homomorphisms(ιn)#m:πm(Fn(S)) →on the πm-level of the long exact sequence in homotopy of the homotopy fibrationas well as an alternative interpretation of Pn(S).One interesting point about this long exact sequence is that the homotopy groups of S2appear,not just as factors in the groupsbut also in the groups πm(Fn(S)).This is a consequence of the following result and the fact that πm(S3)is isomorphic to πm(S2)if m ≥ 3.

Theorem 4.1(see[5,13]and[21,Proposition 6])LetS=S2orRP2,and letn∈N.Then the universal coveringofFn(S)has the homotopy type ofS2ifS=S2andn≤2or ifS=RP2andn=1,and has the homotopy type of the3-sphereS3otherwise.

Let p:S2→ RP2denote the universal covering map,and for i ∈ {1,···,n},let pi:Fn(S)→ S andS denote the respective projections onto the i-th coordinate.Then pi=◦ιn.The following result is proved in[24,Lemma 7 and Proposition 9].

Proposition 4.1(a)Suppose thatS=S2orRP2,and assume thatn,m≥2.Then theinduced homomorphism

(b)Ifn=m=2,then the homomorphism(ι2)#2:π2(F2(S2))is the anti-diagonal homomorphism that sends a generator ofπ2(F2(S2))to(1,−1)or to(−1,1)depending on the choice of orientation ofS2.

(c)Ifn ≥ 2andm ≥ 3,thenis a diagonal homomor-phism.

(d)Letm,n ≥ 2,and suppose thatn=2ifm=2.Then for alli=1,···,n,the homomorphism(pi)#m:πm(Fn(S2))→ πm(S2)is an isomorphism.

(e)Letn,m ≥ 2.Then(ιn)#mis a diagonal homomor-phism,and if additionallym ≥ 3,then(pi)#m:πm(Fn(RP2))→ πm(RP2)is an isomorphism for alli=1,···,n.

If S=S2,in the cases not covered by parts(b)–(d),namely m=2 and n ≥ 3,the homomorphism(ιn)#2:π2(Fn(S2))is trivially diagonal,since π2(Fn(S2))=1,but the homomorphism(pi)#2of part(d)is not an isomorphism.The main ingredients in the proof of Proposition 4.1 are the fact that piis a fibration whose fibre is an Eilenberg-Mac Lane space of type K(π,1),Theorem 4.1,an analysis of ι2and(ι2)#m,and if n ≥ 3,the comparison of the different projections of Fn(S)onto F2(S)given by forgetting all but two coordinates.Proposition 4.1 allows us to determine the homotopy type of the homotopy fibreThe description in the case of RP2makes use of the notion of orbit con figuration space as defined in Section 1 and discussed in Section 2.The following results are stated in[24,Theorem 1].

Theorem 4.2Letn ≥ 2,and letS=S2orRP2.The homotopy fibreIιnof the inclusionmapιn:

Shas the homotopy type ofFn−1(D2),or equivalently ofK(Pn−1,1)×=S2,and has the homotopy type ofequivalently ofK(Gn−1,1)

The idea of the proof of Theorem 4.2 is to replace the map ιnby another map that is null homotopic and whose homotopy fibre has the same homotopy type as that of Iιn.This allows us to apply the general properties of homotopy fibres given at the beginning of this section.The projections piandonto S are fibrations,and the restriction of ιnto their topological fibres gives rise to a mapFurther,the homotopy fibres of ιnandhave the same homotopy type.This is a consequence of the following result that seems to be well known to the experts in the field,but for which we were not able to find a proof in the literature.

Proposition 4.2Letα:E → Bbe a fibration,and letE0be a subspace ofEsuch that the restrictionα0= α|E0:E0→ Bis also a fibration.LetF(resp.F0)denote the fibre ofα(resp.α0)over a base pointb0∈ B,and letι:E0→ Eandι0:F0→ Fdenote the respective inclusions,whereι0= ι|F0.Then the homotopy fibres ofιandι0are homotopy equivalent.

A proof of Proposition 4.2 proposed by Michael Crabb may be found in[24,Proposition 36].In the case where S=S2,by composing by the projections piandwe see that ιn|Fn−1(S2{x0})is null homotopic,which yields the result.If S=RP2,the restriction map ιn|Fn−1(RP2{p(x0)})is not null homotopic,but it lifts to the inclusion mapthat is null homotopic(this may be seen once more by considering the projections onto each factor),and whose homotopy fibre has the same homotopy type as that of ιn|Fn−1(RP2{p(x0)}).One then concludes as in the case of S2.

Recall from Section 2 that Gn=π1(F〈τ〉n(C))and that F〈τ〉n(C)is a K(Gn,1).From the presentation of Gngiven by Proposition 2.2,one may show that the centre of Gnis in finite cyclic,generated by the element Θn= ρ1,0···ρn,0that is similar in nature to Δ2n,and that[24,Remark 21]:

Taking the long exact sequence in homotopy of the homotopy fibration

and using Proposition 4.1 and Theorem 4.2,we obtain the following corollary that may be found in[24,Section 1].

Corollary 4.1Letn≥2.

(a)Letk≥3(resp.k=n=2).Then we have the following split short exact sequence of Abelian groups:

where(ιn)#kis diagonal(resp.anti diagonal).Up to isomorphism,this short exact sequencemay also be written as(b)Ifk=2andn≥3,then we have the following non-split short exact sequence:

which may be also written as1→ Zn→ Pn−1⊕Zn−1→ Pn(S2)→ 1.

(c)Ifk≥3,then we have the following split short exact sequence of Abelian groups:

where the homomorphism(ιn)#kis diagonal.Up to isomorphism,this short exact sequence may also be written as

(d)Ifk=2,then we have the following exact sequence:

which up to isomorphism,may also be written as

In parts(a)and(c),the nature of the exact sequences is completely determined since we know that the homomorphism(ιn)#kis diagonal by Proposition 4.1.In parts(b)and(d),the situation is less clear,and to understand the corresponding exact sequences,one must describe the boundary homomorphismof the homotopy fibration(4.4).This is not so straightforward,since to do so,it is necessary to determine geometric representatives of a generating set of π1(Iιn)via the identifications made in the proof of Theorem 4.2 between the different homotopy fibres,and then relate them to the images under ∂nof geometric representatives of a basis ofThis is quite a long process,and the details are given in[24,Section 5].The basic idea is first to understand the boundary homomorphism in the case n=3(resp.n=2)if S=S2(resp.S=RP2),and then to deduce the result in the general case by considering the projections onto three(resp.two)coordinates.The exact sequences(4.6)and(4.8)may be written in the form:

Note that the termis trivial if S=S2.Fix a base point(x1,···,xn) ∈ Fn(S2).Identifying π2(S)with π1(ΩS)in the usual way,for each 1 ≤ i ≤ n,we choose a generatoriof π2(S2,xi)that is represented by a family of loops based at xi.This gives rise to a basisand to a basis(λp(xi))1≤i≤nUsing the above-mentioned identifications between the various homotopy fibres,if S=S2(resp.S=RP2),we construct an explicit homotopy equivalence h:(resp.h:given by Theorem 4.2.Writing Iιnin the form of Equation(4.1),for i=2,···,n−1,the induced isomorphism h#on the level of fundamental groups sends the generatorof the(i − 1)-th factor ofto an element(resp.δp(xi))of π1(Iιn)whose restriction to the i-th factor of π1(Iιn)is represented byand is constant elsewhere.Further,h#sends(resp. Θn−1)to an element of π1(Iιn)that we denote by τn,and using the geometry,one may see that(pιn)#(τn)=in Pn(S).With our choices of the xiand the corresponding elementsfor i=2,···,n,we prove that(see[24,Theorem 3])

Using these relations and exactness of(4.9),and up to the identification ofwith π1(Iιn),it follows that(see[24,Proposition 4])

By Equations(3.1)–(3.3)and(4.3),it follows that

On the other hand,Ker(ιn#)is equal to Pn(S2)if S=S2,and is equal to Γ2(Pn(RP2))if S=RP2by Proposition 3.1(a)(i),so by exactness of(4.9)and Equation(4.10),we obtain

Equation(4.11)is in agreement with the decomposition given by Equation(3.1).In the case of RP2,we recover the repeated semi-direct product decomposition of Lngiven by Theorem 3.2 by comparing Equation(4.12)with Equation(3.3).

5 Recent Results about the Homotopy Type of the Homotopy Fibre and the Long Exact Sequence,where A is a Con figuration Space and either the Universal Covering of X is Contractible or X is the Orbit Space Sn/G,where G is a Lie Group

Let X be a topological manifold without boundary such that dim(X)≥3.In this section,we describe recent results of[18]about the homotopy type of the homotopy fibre of the inclusion map ιn:and the associated long exact sequence in homotopy.They include the cases where either the universal coveringof X is contractible,or X is an orbit space of the form Sk/G of a tame(i.e.,an action for which the orbit space Sk/G is a manifold,see[8])free action of a Lie group G on the k-dimensional sphere Sk.In the former case,ifeX is contractible,the long exact sequence in homotopy of the homotopy fibration corresponding to ιnis clear.In the latter case,if the group G is finite and k is odd,we will give a full description of the long exact sequence in homotopy of the homotopy fibration of ιn.

To begin with,let X,X′,Y and Y′be topological pointed spaces.The connectivity of X,denoted by conn(X),is defined to be the largest non-negative integer n(or in fi nity)for which πi(X)=0 for all i≤ n.Recall that a map f:X →Y is said to be n-connected if the induced homomorphism f#j: πj(X)→ πj(Y)on the level of πjis surjective if j=n and is an isomorphism for all j≤n−1.Note that the map f is n-connected if and only if its homotopy fibre Ifis(n−1)-connected,in other words,πi(If)=0 for all 0≤i≤n−1.Let f:X →Y,f′:X′→ Y′,g:X → X′and g′:Y → Y′be maps such that the square

is homotopy-commutative.Let H:X ×I → Y′be a homotopy between g′◦f and f′◦g,and let ϕ:X → Y′Ibe the map given by ϕ(x)(t)=H(x,t)for all(x,t) ∈ X × I.Then the map I(g,g′):If→ If′between the homotopy fibres of f and f′defined by

makes the following diagram homotopy-commutative.The two rows are homotopy fibrations,and pf:If→ X and pf′:If′→ X′are the projections onto the first factor described in Section 4.

Now let G×X → X be a tame,free action of a group G on a topological manifold X without boundary.Let q(X):X→X/G denote the quotient map,let q1∈X,let Q={q1},and let=q(X)(Q).The mapX/G induces a ma p ψn(X):(X)→ Fn(X/G),which in turn restricts to a map

Let jn(X/G):denote the maps that insert the point q(X)(q1)into the first position.These two maps are the inclusions of the fibres over the point q(X)(q1)of the fibrations Fn(X/G)→ X andgiven by forgetting all but the first coordinate.By applying the construction of(5.2),we obtain the following diagram that is homotopy commutative:

where the mapsand ιn(X/G)denote inclusion of the given(orbit)configuration space into the associated Cartesian product,the rows are homotopy fibrations,and the mapsand pιn(X/G)are the projections of the homotopy fibres onto the first factor.Using Proposition 4.2,the map

may be seen to be a homotopy equivalence.Applying[18,Corollary 1.7],one may also show that the map

gives rise to a homotopy equivalence

on the level of loop spaces.Let kn(X):be the map defined by kn(X)=(X/G),jn(X/G))◦I(ψn−1(XGQ),q(X)).We then obtain the following theorem.

Theorem 5.1(see[18,Theorem 3.5])LetG×X→Xbe a tame,free action of a Lie groupGon a connected topological manifoldXwithout boundary,and letQ∈X.Suppose thatthe inclusion mapis null homotopic(this is the case if forexample the inclusion mapXGQ → Xis null homotopic).(a)The maps

andare homotopy equivalences,and the map

is a weak homotopy equivalence.Further,ifGis discrete,Xis1-connected anddim(X/G)≥ 3,then the mapψn−1(XGQ):(XGQ)→ Fn−1((X/G)is the universal covering.

(b)Ifj ≤ min(conn(X),dim(X/G)− 2),thenπj((XGQ))=0.

(c)Suppose that the homomorphismπj(X)→ πj(X/G)induced by the quotient mapX →X/Gis injective for allj ≥ 1(this is the case if for example the inclusion mapGQ → Xis null homotopic).For allj ≥ 2,up to the identification of the groupsπj−1(Iιn(X/G))andvia the isomorphism

the restriction to the subgroupof the boundary homomorphismgiven by the long exact sequence in homotopy of the

homotopy fibrationcoincides with the in-clusion ofvia the usual identificationofπj(X)withπj−1(Ω(X)).

The proof of Theorem 5.1 makes use of an explicit model for the homotopy type of the homotopy fibre Ifof a map f:X →Y that is homotopic to a constant map,properties of homotopy pullbacks and universal coverings,as well as a generalisation of[3,Theorem 1]to the topological category given in[18,Theorem 2.2],and routine diagram-chasing arguments.The following corollary is obtained by considering the usual action of π1(X)on the universal covering of X.

Corollary 5.1(see[18,Corollary 3.6])Assume thatXbe a connected topological manifold without boundary such thatdim(X)≥3,letbe its universal covering,and letQ∈.Assume that the map

is null homotopic.

(a)The spacesare homeomorphic,and there existsa homotopy equivalence betweenand a weak homotopyequivalence between

(b)Letj ≥ 2.Up to the identification of the groupsvia theisomorphismand the identification of the groupsπj−1(Iιn(X))andvia the isomorphism given in Theorem5.1(c),the restriction of the boundary homomorphismof the long exactsequence in homotopy of the homotopy fibration

to the subgroupcoincides with the inclusion ofin theproductvia the usual identification ofπj()withπj−1(Ω()).

Corollary 5.1 may be applied to two types of spaces,those for which the universal covering is contractible,and those of the form Sk/G,where G is a Lie group.In the first case,the following result brings together various descriptions of the weak homotopy type of the homotopy fibre Iιn(X),where X is a topological manifold without boundary whose universal covering is contractible.Recall that if f:W →Z and g:Y →Z are maps of spaces,the pullback W×ZY is defined by

Proposition 5.1(see[18,Proposition 3.7])LetXbe a connected,topological manifold without boundary such thatdim(X)≥3and whose universal coveringis contractible,andletQ ∈Then the homotopy fibreIιn(X)of the mapweaklyhomotopy equivalent to each of the following six spaces:

In the remainder of this section,we assume that X is the form Sk/G,where G is a Lie group that acts freely and tamely on Sk,the aim being to generalise the results of Section 4.The following proposition may be proved by applying Theorem 5.1 to the map

Proposition 5.2(see[18,Corollary 4.3])LetG×Sk→Skbe a free,tame action of a compact Lie groupGonSk,and letQ∈Sk.

(a)There is a homotopy equivalence betweenIf the groupGis finite anddim(Sk/G)≥3,then there is a homotopy equivalence between

(b)Ifj ≤ k −dim(G)−2,thenπj

By analysing the long exact sequence in homotopy of the following homotopy fibration

in the case that the group G is finite and k is odd,we obtain short exact sequences similar to those of Corollary 4.1.

Proposition 5.3(see[18,Proposition 4.6])Letk≥3be odd,letj≥2,and letG×Sk→Skbe a free action of a finite groupGonSk.Let(Sk/G)denote the diagonal subgroup ofThen

(a)the image of the homomorphismis equalto(Sk/G);

(b)there are split short exact sequences of the form:

In particular,ifGis the trivial group,then there is a split short exact sequence:

6 Other Types of Con figuration Spaces,and some Open Questions

Let Y be a topological space,let A ⊂ Y be a subspace,and let ι:A → Y denote the inclusion map.The following questions/problems are valid in this general context:

(I)Compute the homotopy groups πi(A)of A.

(II)If j ≥ 0,determine the induced homomorphisms ι#j: πj(A)→ πj(Y).

(III)Describe the homotopy type of the homotopy fibre Iιof the inclusion map ι.

(IV)Determine the long exact sequence in homotopy of the fibration Iι→ Eι→ Y associated to the map ι.

For specific choices of A and Y,it is often more natural to study variations or weaker formulations of these questions.Up until now,we have considered the usual con figuration spaces Fn(X)and Dn(X)(ordered and unordered),as well as the orbit con figuration spaces(X)and(X)(ordered and unordered)with respect to a free action of a group G on a space X.In Section 6.1,we make some comments about the graph con figuration spaces(X)defined in Section 1,which we will refer to as ordered graph con figuration spaces.We discuss some possible quotients of(X)by subgroups of the symmetric group Snthat act freely,and that give rise to some kind of unordered graph con figuration spaces.Finally,in Section 6.2,we propose some questions for the three types of con figuration space discussed in this paper.

6.1 Graph con figuration space

Let Γ be a graph whose vertices are labelled by{1,···,n},that has no loops,and that possesses at most one edge between any two vertices.The n-th graph con figuration space(X)is defined in Equation(1.3)in Section 1.If Γ is the complete graph on the set{1,···,n},i.e.,for all 1≤i＜j≤n,there is an edge in Γ between the vertices i and j,then(X)is the usual ordered con figuration space Fn(X).If on the other hand,the set of edges of Γ is empty,thenis the Cartesian productIn general,we haveLittle is known about these spaces,and the study of the type of questions that we have in mind is at a very early stage.

In order to define a notion of unordered graph con figuration space similar to that for the other types of con figuration space,one needs to adjust the definition given for Dn(X).Recall that the symmetric group Snactsand by restriction,we obtain a free action of Snon Fn(X).Unfortunately,not all elements of Sninduce a map of(X).Indeed,one may check that an element α∈Sninduces a map of(X)if and only if it induces an automorphism of the graph.Let Aut(Γ)denote the group of automorphisms of Γ,that we interpret as a subgroup of Sn.An unordered graph con figuration space could be defined to be the quotient space(X)=(X)/H,where H is a subgroup of Aut(Γ)that acts freely on(X)and is maximal with respect to this property.In principle,with this definition,we may have more than one unordered graph con figuration space because H is not unique in general,and such a phenomenon indeed occurs.Since the quotient map(X)→(X)/H is a covering map,the spaces(X)and(X)/H have the same homotopy groups with the exception of the fundamental group,which coincide rarely.

6.2 Questions and final comments

Among the three types of con figuration spaces described in Section 1 and Section 6.1,the usual con figuration spaces have been studied in much greater detail than the other two types.More is known about orbit con figuration spaces than graph con figuration spaces.For example,at the current time,no analogue of Proposition 2.1 is known for the latter.However,not much is known about the homotopy fibre of the inclusion of the ordered orbit con figuration spaces into the Cartesian product,or about the homotopy groups of the ordered and unordered orbit con figuration spaces,the fundamental group in particular.To end this paper,we state a number of questions about the different types of con figuration spaces that are open(to our knowledge).

(I)For the usual con figuration space Fn(X):If X is a manifold of dimension dim(X)≥3,determine

(a)the homotopy type of the homotopy fibre of the inclusion ιn:

(b)the long exact sequence in homotopy of the fibrationassociated with the map ιn.

(II)For orbit con figuration spaces:If X is a manifold and G is a group that acts freely on X,determine

(a)the homotopy groups of(X),and in particular the fundamental group in the case where X a surface;

(b)the same questions as in(a)for the unordered orbit con figuration space(X)/Sn;

(c)the homotopy type of the homotopy fibre of the inclusion

(d)the long exact sequence in homotopy of the fibrationassociated with the map

(III)For graph con figuration spaces:If X is a manifold and Γ is a graph as in Section 6.1,determine

(a)the surfaces X and graphs Γ for which the space(X)is a K(π,1);

(b)the homotopy groups of(X),and in particular the fundamental group in the case where X is a surface;

(c)the same questions as in(b)for the unordered graph con figuration spaces(X)/H,where H is as defined in Section 6.1.as defined above;

(d)the homotopy type of the homotopy fibre of the inclusion

(e)the long exact sequence in homotopy of the fibrationassociated with the map

We conclude by mentioning that the following two problems constitute work in progress by the authors:

(i)the description of the homotopy type of(S2),and

(ii)with respect to free Z2-actions,the computation of the pure orbit braid groups π1((X)),where X is S2,the torus or the Klein bottle,and of the full orbit braid groups π1((X)/Sn),where X is the cylinder C,S2,the torus and the Klein bottle.

AcknowledgementsThis work started during the visit of the second author to the Departamento de Matem´atica do IME–Universidade de S˜ao Paulo(Brazil)during the period 19thOctober–3rdNovember 2015.Many of the ideas for this paper originated during the Combinatorial and Toric Homotopy Conference in honour of Frederick Cohen that took place in Singapore in August 2015.We are indebted to the organisers of this meeting,Professors Haibao Duan and Zhi L¨u,for their encouragement to write this paper for the proceedings of the conference.

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