arXiv:1710.07116v1 [math.DG] 19 Oct 2017
Quasifolds
Elisa Prato
Dipartimento di Matematica e Informatica ”Ulisse Dini” Universit`a degli Studi di Firenze
Piazza Ghiberti 27 50122 Firenze, ITALY elisa.prato@unifi.it
Quasifolds are singular spaces that generalize manifolds and orbifolds. They are locally modeled by manifolds modulo the smooth action of countable groups and they are typically not Hausdorff. If the countable groups happen to be all finite, then quasifolds are orbifolds and if they happen to be all equal to the identity, they are manifolds. For the formal definition and basic properties of quasifolds we refer the reader to [20, 6]. In this article we would like to illustrate quasifolds by describing a 2–dimensional example that displays all of their main characteristics: the quasisphere. The reader will not be surprised to discover that quasispheres are generalizations of spheres and orbispheres, so we will begin by recalling some relevant facts on the latter two.
From sphere to orbisphere to quasisphere
The sphere
Let us write the 2 and 3–dimensional unit spheres as follows S2 = { (z, x) ∈ C × R | |z|2+ x2 = 1 },
S3 = { (z, w) ∈ C2| |z|2+ |w|2 = 1 }. The surjective mapping
f : S3 −→ S2
(z, w) 7−→2zw,|z|2− |w|2
is known as the Hopf fibration. It is easily seen that the fibers of this mapping are given by the orbits of the circle group
S1 = { e2πiθ| θ ∈ R } acting on S3 as follows:
Therefore S2 can be identified with the space of orbits S3/S1. Notice that the S1-orbits through the points (0, 1) and (1, 0) of S3 correspond, respectively, to the south pole,
S = (0, −1), and north pole, N = (0, 1), of S2.
The orbisphere
This simple quotient construction can be extended to the orbifold setting as follows. Let p, q be two relatively prime positive integers and consider the 3–dimensional ellipsoid
Sp,q3 = { (z, w) ∈ C2| p|z|2+ q|w|2= pq }. The circle group S1 acts on S3
p,q as follows:
e2πiθ· (z, w) =e2πipθz, e2πiqθw. (1) Taking the space of orbits in this case yields the 2–dimensional orbifold Sp,q2 = Sp,q3 /S1, called orbisphere. It admits the two singular points S = [0 : √p] and N = [√q : 0]. We will come back to these singularities later. From the complex algebraic point of view, this orbisphere is isomorphic to a weighted projective space CP1p.q (see Holm [16] for this and more on orbispheres).
The quasisphere
We now extend the construction even further. Let s, t be two positive real numbers with s/t /∈ Q and consider the 3–dimensional ellipsoid
Ss,t3 = { (z, w) ∈ C2| s|z|2+ t|w|2= st }.
We would be naturally tempted to replace p, q with s, t in (1), but this does not define an S1–action on S3
s,t: in fact, if you replace θ by θ + h, where h is a non–zero integer,
we have e2πi(θ+h) = e2πiθ but (e2πis(θ+h), e2πit(θ+h)) 6= (e2πisθ, e2πitθ). However, if you replace the circle S1 with R:
θ · (z, w) =e2πisθz, e2πitθw, θ ∈ R.
we get a well–defined action. We define our 2–dimensional quasisphere to be the space of orbits Ss,t2 = Ss,t3 /R. This quotient is the simplest example of quasifold. It is wilder then the sphere and orbisphere, in that it is not a Hausdorff topological space. However, surprisingly, quasiphere charts are a straightforward and very natural generalization of standard sphere and orbisphere charts. We will show this in the following section.
Charts
In this section, for any positive real number r, we will denote by B(r) the open ball in the space C of center the origin and radius √r. Moreover, for each (z, w) ∈ S3, S3
p,q or
Sphere charts
Consider the covering of S3/S1 given by the open subsets US = { [z : w] ∈ S3/S1 | w 6= 0 }
UN = { [z : w] ∈ S3/S1| z 6= 0 }.
As the notation suggests, the first is a neighborhood of the south pole S = [0 : 1], while the second is a neighborhood of the north pole N = [1 : 0]. They are charts because of the homeomorphisms: φS: B(1) −→ US z 7−→ z :q1 − |z|2 φN: B(1) −→ UN w 7−→ q 1 − |w|2 : w . Orbisphere charts
Similarly for the orbisphere, consider the covering of S2
p,qgiven by the two open subsets
US = { [z : w] ∈ Sp,q3 /S1| w 6= 0 }
UN = { [z : w] ∈ Sp,q3 /S1 | z 6= 0 }.
The first is a neighborhood of the point S = [0 : √p], while the second is a neighborhood of the point N = [√q : 0]. Let us show that they are are each homeomorphic to the quotient of an open subset of C modulo the action of a finite group. This will imply that they are 2–dimensional orbifold charts. The group Zq acts on the open ball B(q)
by the rule (k, z) 7→ e2πikq · z, k = 0, . . . , q − 1. Consider the orbit space B(q)/Z
q and,
for any z ∈ B(q), denote by [z] the corresponding orbit. The mapping φS: B(q)/Zq −→ US [z] 7−→ z : r p −pq|z|2
is a homeomorphism. Similarly, the group Zp acts on the open ball B(p) by the rule
(m, w) 7→ e2πimp · w, m = 0, . . . , p − 1, and the mapping
φN: B(p)/Zp −→ UN
[w] 7−→
r
q − qp|w|2 : w
Quasisphere charts
Exactly as above, we consider the covering of S2s,t given by the opens subsets US = { [z : w] ∈ Ss,t3 /R | w 6= 0 }
UN = { [z : w] ∈ Ss,t3 /R | z 6= 0 }.
The first is a neighborhood of the point S = [0 :√s] while the second is a neighborhood of the point N = [√t : 0]. Let us show that they are are each homeomorphic to the quotient of an open subset of C modulo the action of a countable group. This will imply that they are 2–dimensional quasifold charts. The group Z acts on the open ball B(s) by the rule (k, z) 7→ e2πikst · z. Consider the orbit space B(s)/Z and, for any z ∈ B(s),
denote by [z] the corresponding orbit. The mapping φS: B(t)/Z −→ US [z] 7−→ z : r s − s t|z| 2
is a homeomorphism. Similarly, the group Z acts on the open ball B(t) by the rule (m, w) 7→ e2πimst · w and the mapping
φN: B(s)/Z −→ UN
[w] 7−→
"r
t − ts|w|2 : w
#
is also a homeomorphism. Therefore the opens subsets US and UN are quasifold charts.
We conclude by remarking that the quasifold change of charts is given by gSN = φ−1N ◦ φS: φ−1S (US∩ UN) −→ φ−1N (US∩ UN) [z] 7−→ " z |z| st r s −s t|z| 2 # .
History and context
We end this brief account on quasifolds with a few comments on history and context. Quasifolds were initially introduced in [19, 20] in order to make sense of symplectic toric geometry for those simple, convex polytopes that are not rational. Rationality is a very strong condition, and in fact many of the polytopes that come to mind are not rational: the regular pentagon, the Penrose kite and the regular dodecahedron, just to mention a few.
The main idea in [19, 20] was to generalize the Delzant construction [15], which allowed to associate a symplectic toric manifold with each simple, rational, smooth, convex polytope. A first generalization of this construction was actually given by Lerman–Tolman [18]: they dropped the smoothness condition on the polytope at the expense of allowing orbifold singularities on the toric space. Dropping the rationality
condition, however, is more drastic and this is where quasifold singularities came into play. One of the key ideas was to replace a lattice in Rn with the Z–span of a set of
R–spanning vectors. Incidentally, the latter arises also in the context of quasicrystal geometry (see Senechal [23, Chapter 2]) and is sometimes referred to as quasilattice. It was actually this connection to quasicrystals that inspired us to call these singular spaces quasifolds. It has also motivated us to explore, in joint work with Battaglia, Penrose rhombus tilings [5] and Ammann tilings [7] from the viewpoint of symplec-tic geometry. A symplecsymplec-tic toric quasifold associated with the Penrose kite, on the other hand, was described, again jointly with Battaglia, in [6]; the case of the regular dodecahedron was treated in [21].
A complex counterpart of the non–rational toric construction was given, jointly with Battaglia, in [4]. A special mention goes to Battaglia for generalizing these con-structions to non–simple convex polytopes, both in the symplectic [1] and complex [3] category. Her work can be applied to the regular octahedron and the regular icosahe-dron, for example, since they are both not simple. A full account of the toric spaces associated to the five regular convex polyhedra can be found in [9]. Other developments of the theory of quasifolds can be found in [2, 8, 10].
We conclude by mentioning that toric quasifolds have been viewed as leaf spaces by Battaglia–Zaffran [11, 12, 13]. Quasifold–type singularities could also be studied from the viewpoint of non–commutative geometry [14, Chapter II] and of diffeology [17]. Ratiu–Zung, on the other hand, have recently suggested an alternate non–rational toric approach via presymplectic geometry [22]. How all of these different mathematical paths connect precisely, is still a matter of debate.
References
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(2012), 39 pages.
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