The Euler class of planar groups
This is an exposition of the homological classification of actions of surface groups on the plane, in every degree of smoothness.
This note gives an exposition of Theorems C and D from the paper . In particular, it gives the homological classification of actions of orientation-preserving closed orientable surface groups on the plane, in every degree of smoothness. I would like to the thank the referee for reading the paper carefully, and making useful corrections.
2. Euler class
2.1. Homotopy type
Let denote the group of orientation preserving homeomorphisms of the plane. This is a topological group with the compact-open topology.
The group with the compact-open topology is homotopy equivalent to . Consequently there are homotopy equivalences of classifying spaces
where in dimension generates a free polynomial algebra. The element is usually known as the Euler class.
One suggestive way to see this homotopy equivalence is to think of as . Given an element of there is a unique , the affine group of , such that fixes and therefore induces a map from the thrice-punctured sphere to itself. If is sufficiently regular, for instance, if it is quasiconformal, then there is a Beltrami differential such that
If with denotes the unique quasiconformal self-homeomorphism of fixing with Beltrami differential , then defines a homotopy in from to id. This demonstrates that the coset space of in the subgroup of quasiconformal transformations in is contractible, and therefore that this subgroup at least has the homotopy type of , which is to say, of a circle. Classical arguments can be adapted to show that the inclusion of the group of quasiconformal homeomorphisms into the group of all homeomorphisms is a homotopy equivalence (compare with Kneser , § 2.2 and § 2.3).
Given a group , an action of on by orientation preserving homeomorphisms is the same thing as a homomorphism
We can pull back by to an element in group cohomology .
Let denote the closed, orientable surface of genus . In the case the class can be evaluated on the fundamental class which generates (implicitly we need to choose an orientation on to define ). We call this value the Euler number of the action.
The basic question arises as to what Euler numbers can arise for different , and for different constraints on the analytic quality of .
The group also has the homotopy type of , and therefore we may ask an analogous question for actions of surface groups on circles. In this case, the well-known Milnor-Wood inequality is the following inequality.
Moreover it is known that every value which satisfies this inequality can be realized by an action with any degree of smoothness (in fact, even for analytic actions).
In this section we give several methods for computing .
2.2.1. Central extension
For a group , elements of classify central extensions of , i.e. short exact sequences where is central, up to isomorphism. Let denote the group of germs of elements of at infinity. Every neighborhood of infinity may be restricted further to some annulus neighborhood . If denotes the universal cover of , then elements of can be lifted to germs of homeomorphisms of . Let denote the group of lifts of this form. Then there is a central extension
where acts on as the deck group. The class of this extension is .
Note from this construction that the Euler class depends only on the germ of a group action at infinity. Another way to see this fact is as follows. Let denote the subgroup of consisting of homeomorphisms of compact support. Then is obviously normal, and there is an exact sequence
Moreover, is contractible, as can be seen by the Alexander trick: let be the dilation . Then for any the family as goes from to is a path in from to id. This construction shows that is contractible, and therefore that comes from a class in the cohomology of .
Given one can form a bundle
over with fiber . The total space of is a (noncompact) oriented -manifold. Since the fiber is contractible, there is a section . The Euler class of the bundle is the self-intersection number of this section
If the action of is differentiable, the following related construction makes sense. Let be an equivariant developing map. Since is contractible, such a developing map can be constructed skeleton by skeleton over a fundamental domain for . The vector bundle pulls back to an (vector space) bundle over . Since by hypothesis the action of is differentiable, it acts on by bundle maps, and therefore also on . The quotient is an (vector space) bundle over , and is the obstruction to finding a non-zero section of this bundle.
These two bundles are closely related. The developing map determines and is determined by the section . If the action is smooth, can be chosen to be smooth, and is the normal bundle in of .
2.2.3. Graphical formula
Suppose the action of is at least . Let be a fundamental polygon for and the image of the boundary under a developing map which is an immersion on each edge of . Let be obtained from by “smoothing” the image at the corners; i.e. is the image of the boundary of a regular neighborhood of in . Then there is a formula
where denotes the winding number of .
To relate this to the previous definition, observe that the immersion defines a trivialization of over the -skeleton, with a multi-saddle singularity at the vertex. The winding number of represents the obstruction to extending this “trivialization” over the -skeleton.
The following example seems to have been first observed by Bestvina.
Example 2.5 (Bestvina).
Let be a Dehn twist in a thin annulus centered at the origin with radius approximately . Let be the dilation , and suppose that the closure of is disjoint from its translates under powers of . Define
In other words, is the product of Dehn twists in a family of concentric annuli all nested about the origin. By construction , so there is a representation taking one generator to and the other to . Observe that is only at the origin, but that and are away from the origin.
Notice that and both fix so the Euler class represents the obstruction to lifting to an action on the universal cover of . Let and let denote this cover. If we think of as , then we can take and the covering map can be taken to be the exponential map. A natural lift of is . However, if is a lift of which fixes some for which is not in the support of , then acts on the line (for fixed integer and for ) as . It follows that is the translation , which is the generator of the deck group of the covering . In other words, the value of the Euler class on is .
Replacing by produces an action of with Euler number . Factoring this action with a degree one map gives an action of on with any Euler number.
In other words:
For any and any there is a representation with .
Consider the following construction.
Let and be as in Example 2.5. Define
It follows that . Let be a homeomorphism with for and for , and define
Then so the four elements together define a representation .
The image of in is trivial, so the Euler class breaks up into a contribution from two copies of in . The first action has the same image in as the action described in Example 2.5. The group generated by and fixes all points with sufficiently negative, and such fixed points can be lifted to give a section of the image in to , so the Euler class vanishes on this second copy of . Hence in this example. If is replaced by then .
For any and any there is a representation with .
It remains to understand what Euler numbers may be realized by actions of which are or smoother. We address this in the next section.
3. actions of
The main purpose of this section is to prove:
Let be a action of on the plane. Then .
The proof of this theorem divides up into three cases, depending on the dynamics of the action, and the proof in each case depends on constructions and techniques particular to that context. We do not know of a single approach which treats all of the cases on a unified footing. The cases are as follows:
One generator (call it ) fixes , and the orbit of under powers of the other generator (call it ) is not proper
fixes , and the orbit of under is proper
The action is free (i.e. no nontrivial element has a fixed point)
These cases are treated in subsequent sections.
3.1. fixes , the orbit of under is not proper
The simplest case is that and (some power of) have a common fixed point. Since the Euler class is multiplicative under coverings, we can assume and have a common fixed point. Projectivizing the induced action on the tangent space at this point gives an action on the circle . By the Milnor-Wood inequality (Theorem 2.2) The Euler class of this action vanishes.
Slightly more complicated is the case that fixes some point , and there are integers so that converges to some point . Since and commute, it follows that fixes each , and therefore also .
To analyze this case we introduce some technology which will be useful in what follows.
Given distinct points and an isomorphism , define to be the space of embeddings such that and , and satisfying .
If are understood, we abbreviate by .
Lemma 3.3 (Whitney).
The set of path components is an affine space for , where acts on by a positive Dehn twist in a small annulus centered at the positive endpoint.
This operation is illustrated in Figure 1.
Given an element the class of in , denoted , is called the writhe of . Since is an affine space for , given elements the difference of writhes is well-defined.
Let be standard generators for acting on at least by . Let . Let be arbitrary. Then .
A -parameter family of smooth curves from to pulls back to define a trivialization of over a fundamental domain. The obstruction to this trivialization is . If then there is a family from to which differs from only by twists at one endpoint. This family defines a trivialization of with a singularity of order at one point. ∎
This also follows from the graphical formula for the Euler class, described in § 2.2.3.
In our context, we have fixed points of which are very close to . Let be a smooth curve from to with . So . Since is , if are close enough, the action of near is close to a linear action, so and are close. In particular, and therefore the Euler class is zero on the fundamental class of the group generated by and . But this is times , which must therefore vanish.
3.2. fixes , the orbit of under is proper
Let , so that , by Lemma 3.4. Choose such that does not intersect for any except at the endpoints and . Since the orbit is proper (and in any case, countable) this is easy to achieve.
For each integer , we define by the formula
where denotes intersection number. For the endpoints of or and of are disjoint, so this is well-defined. For we must be careful, since a priori and might have infinitely many points of intersection near , and similarly for and . We replace with which agrees with outside of a small neighborhood of , which agrees with in a smaller neighborhood of , and which satisfies . Notice that for if and agree outside a neighborhood of that does not contain any with .
For sufficiently large we have .
The union (i.e. with the opposite orientation) is a closed, oriented loop. Since the orbit is proper, for sufficiently large the points are outside a big disk containing and therefore
The next Lemma gives a formula for the Euler class in terms of the , thus relating the Euler class to the dynamics of the action.
Note by Lemma 3.6 that the sum on the right hand side is finite, and therefore well-defined. By Lemma 3.4, the left hand side is equal to the difference in writhes . Set (as in the definition of the above) and let denote a (suitable) -parameter family of curves in which are constant in a sufficiently small neighborhood of and . For each which does not pass through any we can define and therefore the sum .
Since the orbit is proper, there are only finitely many values of when a generic family passes through some . Since each is embedded, it can only pass through where . When it passes through such a point, changes by and changes by , so stays constant.
After a suitable deformation, we can assume agrees with except in a small annulus neighborhood of (or ) where it differs by the th power of a Dehn twist, where . It follows that for all , and . Hence
as required. ∎
The proof in the second case is completed by a covering trick. Fix some large integer , and define . Replace by and by . Define coefficients . The group generated by and is index in the group generated by and . It follows from Lemma 3.7 that there is a formula
However, it is also true that
Part of the graph of is depicted in Figure 2.
Notice that for any there is an equality and therefore
which is a finite sum, by Lemma 3.6. In other words, the right hand side is eventually constant, and therefore the left hand side is identically zero, as we needed to show.
3.3. The action is free
In this case we must cite Theorems of Brouwer and Brown.
Theorem 3.8 (Brouwer ).
If is fixed-point free, the orbit of every point is proper. Moreover, for any point there is an arc from to such that the union of the translates is an embedded copy of in .
The caveat to Theorem 3.8 is that the copy of obtained may not be properly embedded.
Let be fixed-point free. An embedded arc from to with the property that is an embedded copy of is called a free arc.
Brown gave a useful criterion for an arc to be free.
Theorem 3.10 (Brown ).
If is fixed-point free, and is an embedded arc from to for which then is a free arc.
If is fixed-point free and , then for any point , one can choose a free arc in . The next Lemma shows that the writhe of a free arc is well-defined.
Let be free. Then .
For an element define
leaving aside for the moment whether this sum is finite. If is free, then of course , since the are termwise zero. Since is free, the orbit is proper by Theorem 3.8. If is a -parameter family in , the value of only changes when crosses over some , where necessarily since elements of are embedded.
At such a crossing, four terms in the sum change: changes by , changes by , changes by and changes by . It follows that is constant (if it is defined at all) on classes in .
If we change by performing a small positive Dehn twist at , it changes the writhe by . It also changes by and by , so changes by . It follows that there are canonical co-ordinates on when is free, determined by the formula
In particular, . ∎
We now return to our original problem. Pick a point , and let be a smooth arc from to . For each let . As varies in , this determines a bundle over which we denote . Since the spaces vary continuously as varies, there is an induced flat connection on of the fibers. Since the base space is an interval, this gives us a canonical identification for any two .
By Theorem 3.8 and Lemma 3.11, every contains a distinguished element, which contains all the free arcs. By Theorem 3.10, free arcs are stable under perturbation, so the set of free classes defines a continuous section of the bundle. In particular, the canonical identification between of fibers identifies the free class in each fiber.
The element induces a map by for . If is free, then so is , so their respective classes in and are equal under the canonical identification. It follows that we can choose a continuous section with and . The family of tangents to this family of smooth curves pulls back by to give a nonvanishing section of , and therefore .
This completes the proof of Theorem 3.1.
-  L. E. J. Brouwer, Remark on the plane translation theorem, Nederl. Akad. Wetensch. Proc. 21 (1919) 935–936
-  M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math. 11 (1985), 455–469
-  D. Calegari, Circular groups, planar groups, and the Euler class, Geom. Topol. Mon. 7 (2004), 431–491
-  H. Kneser, Deformationssätze der einfach zusammenhängenden Flächen, Math. Zeit. 25 (1926), 362–372
-  J. Milnor, On the existence of a connection with curvature zero, Comm. Math. Helv. 32 (1958), 215–223
-  J. Wood, Bundles with totally disconnected structure group, Comm. Math. Helv. 46 (1971), 257–273