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The Con Guration Space of the Three Graduate J. Math. 2 (2017), 29 { 36 The conguration space of the three dimensional Lens space L(7; 2) and its model Giulio Calimici Abstract obtained as a suitable quotient of the three sphere S3 (denitions in the text). This Lens space; which is We study the algebraic topology of the two point conguration one in a family of spaces with relatively space on the lens space L(7; 2), describing an explicit nite dimensional L(p; q) p; q model for the algebra of De Rham dierential forms on the universal prime, was used by Longoni and Salvatore [13] to cover of the two point congurations space Ω∗ (F~ (L(7; 2))). provide the only known counterexample to the con- DR 2 jecture that the diagonal complement of a closed ori- MSC 2010. Primary 55R80; Secondary 55S30. ented manifold, or equivalently its two points con- guration space, is a homotopy invariant. In con- structing the counterexample, one takes advantage of the fact that some Lens spaces are known to be 1 Introduction homotopy equivalent but not homeomorphic. Note that the diculty in dealing with this conjecture is Rational (or real) homotopy theory provides a rare that plain homological or homotopy group calcula- instance in topology where spaces, which one takes to tions cannot distinguish between the various diagonal be simply connected, can be completely understood complements associated to L(p; q). There is a need by an algebraic model once we only consider the non- to look for deeper invariants, like Massey products, torsion invariants of the space. The topological dic- and thus the need to go through universal covers to tionary and the algebraic dictionary are completely detect those subtle dierences, as was done in the equivalent in this case. This theory is vast and we Longoni-Salvatore paper. only refer to the main reference on the subject [7]. Denote by ~ the universal cover of the To better explain what we do in this paper, we F2(L(7; 2)) Lens space F2(L(7; 2)). Our main results are given must say more about these algebraic models which in section 3 where the model is constructed and then always come in the form of commutative graded dif- in section 4 where we furthermore observe that the ferential algebras, or CGDAs for short. The dier- model is equivariant with respect to the action of the ential is denoted by d. For a smooth dierential fundamental group. As an application of our con- manifold , the deRham dierential forms ∗ M ΩDR(M) struction, we recover the non-trivial massey product form a CDGA over the real numbers and this deter- in the cohomology of F2(L(7; 2)) discovered by Lon- mines the real homotopy type of M (in particular, goni and Salvatore. the ranks of the homotopy groups). This CDGA is however generally very large and one seeks to nd a smaller more manageable CDGA quasi-isomorphic to Acknowledgements: This work was part of the master thesis Spazi di congurazione delle varietá Ω∗ (M), thus yielding the same real homotopy type. \ DRMore precisely, a CDGA is called a model for lenticolari from the academic year 2014/2015 of the A author at the University of Roma Tor Vergata under Ω∗ (M) if there is a third intermediate CDGA B mappingDR homomorphically to both the guidance of Prof. Paolo Salvatore. ∗ A − B −! ΩDR(M) 1.1 Conguration spaces with the arrows inducing an isomorphism in coho- The conguration space of n points in a manifold mology. All algebras in our case are dened over R. M is the space Our goal in this paper is to exhibit a nite di- mensional model for the De Rham dierential ×n (A; d) Fn(M) := f(x1; : : : ; xn) 2 M : xi 6= xj for i 6= jg: forms on the universal cover of a particular three di- (1.1) mensional manifold which is the Lens space L(7; 2), 29 2 Two point conguration space of lens spaces 30 or equivalently the complement Denition 1.2. Let (p; q) 2 Z be relatively prime. n The lens space L(p; q) is dened to be the quotient Fn(M) = M − ∆; 3 S =hζqi. where ∆ denotes the diagonal Observation 1.3. The projection S3 ! L(p; q) is for some a covering space projection since the action of is ∆ = f(x1; : : : ; xn)j xi = xj i 6= jg: Zp free. Only the identity element has xed points in One thinks of as all -tuples of points of Fn(M) n M 3, and for each point 3, 2πikq=p , for all that are pairwise distinct. S zj 2 S e zj 6= zj Conguration spaces have been studied in many 0 < k < p. This is a consequence of the assumption that . dierent situations. For example, 3 is the nat- mcd(p; q) = 1 Fn(R ) ural setting for the -body problem (for a history n Because S3 is simply connected, it follows immedi- 3 of this topic see [4]). Congurations of points in R ately that S3 is the universal covering of L(p; q) and that are required to adapt to a given geometry are that ∼ . An important theorem of Rei- π1(L(p; q)) = Zp used in robotics and the piano mover's problem (for demeister [18] classies the Lens spaces up to home- more details on this topic, see [5]). In mathematical omorphism and homotopy equivalence. physics, conguration spaces have been considered in relation to gauge theory, gravity or particle physics. Theorem 1.4. The lens spaces L(p; q) and L(p; q0) In topology and geometry, these spaces enter as a are homotopy equivalent if and only if for some m 2 we have that: fundamental tool in the study of spaces of functions Zp and moduli spaces. 0 2 Exemple 1.1. The best-known and simplest non q ≡ ±qm mod p: trivial examples are the conguration spaces 2 , Fn(R ) They are homeomorphic if and only if whose fundamental groups 2 are the Pn = π1(Fn(R )) so-called pure braid groups. The classical Artin braid q0 ≡ ±q±1 mod p: groups appear as the fundamental groups of the unordered conguration spaces 2 , Bn = π1(UFn(R )) where UF ( 2) is dened to be the quotient of F ( 2) 2 Two point conguration space of lens n R n R spaces by the natural action of the symmetric group Sn 2 which permutes the points in R . The braid descrip- tion is used to visualize paths in the conguration In this section we will discuss the cohomology of space, thus providing a very intuitive description. the universal cover of two point conguration spaces of lens spaces, as calculated in [13], and we present a list of generators of the relative homology groups 1.2 Lens Spaces 3 3 H∗(S × S ; ∆q), where ∆q is dened below, and de- The term \lens space usually refers to a specic scribe their respective Lefshetz duals. class of 3-manifolds, albeit these can be dened in The universal cover of the two point conguration higher dimensions. They were introduced for the space F2(L(p; q)) is given as follows rst time by H. Tietze in 1908. There is more than ~ 3 3 k one way to construct lens spaces in the 3-dimensional F2(L(p; q)) = f(x; y) 2 S ×S : x 6= ζq y 8 k 2 Zpg: case. One way is to take the quotient of the unit 3- sphere S3 by an action of a cyclic group. Another For each q = 0; : : : ; p − 1, dene way is by gluing two solid tori together via a home- [ k omorphism of their boundary. ∆q := ∆q In the rest of the paper we identify to the group Zp k2Zp of pth complex roots of unity generated by where ∆k is the image of the following embedding of 2πi=p q ζ = e : S3 3 3 3 Let S3 the unit 3-sphere viewed as a submanifold of S −−!S × S 2; k C ! 7−! (!; ζq !): 3 ∼ 2 2 2 S = f(z1; z2) 2 C j jz1j + jz2j = 1g: If we write ! = (x1; x2) in complex coordinates, then k k kq Dene for each relatively prime to dene (!; ζ !) = ((x1; x2); (ζ x1; ζ x2)), and we can view q 2 Zp p q the action of on 3 such that the generator acts ~ as the subset of 2 2 given as follows Zp S F2(L(p; q)) C × C by the map ζq with k qk q f((z1; z2); (z3; z4)) j (z1; z2) 6= (ζ z3; ζ z4); k 2 Zpg: ζq(z1; z2) = (ζz1; ζ z2) where we denote by z1 and z2 the complex coordi- nates of S3. 2 Two point conguration space of lens spaces 31 ¯ 2.1 Representatives in homology and duality closure of the manifold Ak denoted by Ak. The long exact sequence restricts to From now on we denote I(k) the open interval (k − 1; k) and x the positive orientation of I(k) as 3 3 the canonical one.We also choose the orientation of H4(S × S ) = 0 S3 to correspond to the orientation at the point (0; 1) given by the basis tangent vectors (i; 0); (1; 0) and @ (0; i). In [13], that authors computed the following H (S3 × S3; ∆ ) / H (∆ ) / H (S3 × S3) ··· cohomology groups 4 q 3 q 3 ¯ k k−1 8 6 @ :[Ak] 7−! [∆q ] − [∆q ]: <R ∗ = 2 ∗ ~ the algebraic boundary is deduced from the geo- H (F2(L(p; q))) = R ∗ = 3 @ 6 metric boundary of ¯ which is exactly k k−1.
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