The smooth fundamental bigroupoid is a smooth version of the topological fundamental bigroupoid, constructed essentially by replacing all continuous maps with smooth ones. This introduces some technicalities, mainly relating to the fact that one can no longer concatentation smooth maps merely if they match up on their common boundary.
First, let us define the points of our space. Let us fix once and for all a manifold (without boundary). We start with “homotopies of paths”:
A homotopy of paths in is a smooth map such that and are both constant paths.
The source of is the path and the target is the path .
By a “smooth map” with source , we require all derivatives to extend (continuously) to the boundary. Equivalently, the smooth map comes from a morphism in the category of Froelicher spaces. We could express this as simply a smooth map from the quotient of by the two lines, but that is not the same as the space obtained by imagining a “bi-gon” in .
Now we want to establish our equivalence relation.
We define an equivalence relation on the set of homotopies of paths in according to the following: if there is a smooth map such that:
Of course, we need to establish the fact that the above is an equivalence relation. Reflexivity and symmetry are obvious. For transitivity, we need to adjoin two such maps together. We do so by placing the two copies of one on top of the other and defining a map by where is increasing, surjective, and is flat at .
A 2-track is an equivalence class of homotopies of paths under the above relation. We write for the set of all -tracks in .
Now we want to put the structure of a manifold on the space of 2-tracks. We do so following the structure of that of a smooth loop space.
Let be a local addition. Let be a 2-track. As such, it has well-defined source and target paths, say and . These are smooth paths in with common end-points. As such, they can be considered to be a smooth map from the smooth space formed by taking two copies of the interval and identifying the endpoints (note that this is not the circle). Let us denote this space by . Let us write for the map which is on, say, the left half and on the right half.
We then write for the space of smooth sections of the pull-back of the tangent bundle of by . We should explain exactly what this means. This is the space of smooth maps covering . Thus it is the subspace of consisting of sections over each piece which agree at the end-points. Thus, in particular, the only requirement at the end-points is that the map be continuous, not smooth.
We now define . Let us start with a section, . As this is a section, it defines a map . We compose this with the local addition, , to define a map . Now we need to fill this in to a -track. We do this by using the diffeomorphisms associated to the local addition.
The local addition defines a smooth map . Thus, composing with , we have a smooth map (and note that this varies smoothly with ). This has the property that is always the identity on and . Thus moves the point to the point , dragging the rest of along after it like a rubber sheet. So by applying , we can distort so that maps to (this picture is not quite right, since the distortion is parametrised by the points of ). The trick is now to drag along with us.
To do this, we need to define a “weight” function on which says how much of the distortion a particular point in feels, and which bit of the boundary of is providing the distortion. More concretely, we want to define a function which is on the boundary and we want to define a function with certain properties. We shall then define the 2-track as
So we use to tell us which distortion to use and to tell us how much of that distortion to apply. Then we apply this distortion to .
The properties that we want to have tell us what conditions we need on and . We want to be smooth and we want to restrict to on the boundary (via the collapse map ). We therefore need to induce the same collapse map on , thus it must preserve two of the sides pointwise and preserve the other two in totality. This means that cannot be defined on the whole of . Therefore must be a bump function on the domain of such that takes the value on .
Let us give a semi-explicit definition of . First, we choose a smooth function which is increasing and maps to and to . We also choose a smooth function which is zero on and and is on . Then we define
Let us check what happens here. When then is either or so . Moreover, if then and so the first co-ordinate of is just . When then and so the first co-ordinate of is . Thus if then . Hence takes the square annulus to and has the desired properties on the boundary. So we restrict to this annulus and choose an appropriate bump function .
This then defines our -track , and thus our chart map on . That the transition functions are diffeomorphisms will follow by a similar argument to that at loop space.
This argument generalises somewhat. We retain the background space as a smooth manifold, , but generalise the source to a sequentially compact Frölicher space, . We formally declare to have a “boundary”, . By “formally declare” we are not assuming the existence of a natural boundary of but simply assert that part of the initial data is a subset of , satisfying some conditions, that we call its boundary. The conditions that we require are as follows:
With these assumptions, we can follow the above, replacing one copy of by on all occasions. We start by defining our homotopies; these are smooth deformations of maps from to which do not deform . Precisely, we make the following definition.
A homotopy of -maps is a smooth map such that the induced map is independent of the -parameter.
From this, we define our equivalence relation.
We define an equivalence relation on the set of -maps according to the following rule: if there is a smooth map such that:
That this is an equivalence relation follows exactly as before since the argument uses a smooth reparametrisation in the new -direction.
An -track is an equivalence class of homotopies of -maps under the above relation. We write for the set of all -tracks in .
The notation is almost the same as that above. If we label as (which is a reasonable thing to do!) then we are renaming a -track as an -track.
We now want to put a manifold structure on in the same manner as above. So we fix a local addition on and choose an -track, . This has well-defined source and target paths, obtained by restricting a choice of map for to . These are smooth maps which agree on . As such, they define a smooth map .
Our model space is then , smooth sections of the pullback of along . A point in this space is a smooth map such that . We compose with the local addition to obtain a map . Now we need to fill this in.
Let us choose a representative for , and label it again .
The local addition defines a smooth map . Thus composing with we obtain a smooth map (varying smoothly with ). This has the property that is always the identity on and . Thus is a path from to and is a deformation of that drags along that path.
To drag the whole of along with us, we need a “weight” function on . That is, we need a function which is on , and we need a function . We shall define as
Thus tells us which distortion to use, tells us how much of that distortion to use, and we then apply that to .
The properties that we want to have tell us what conditions we need on and . We want to be smooth and we want it to restrict to on the boundary (via the obvious collapse map ). We therefore require to satisfy
Since we have more restrictions in the -direction, we start by ensuring that the first condition is met. So choose a smooth function which is increasing and maps to and to . Let be a smooth function which is zero on and and is on . Let be a smooth function as in the assumptions on . Define
Let us check what happens here. When then is either or so . Then when (as in the conditions on ) and , so . Hence maps the open set into .
Now, if then and so , whence for . Then if , for all so for .
The function must be a bump function taking value on and with support on into . By assumption, there is a smooth function such that and . Let be a smooth function such that and . Define by:
Then if , and if , . But if then and so . Hence is the required bump function.
Thus we can define our chart map as
Last revised on June 9, 2010 at 08:23:55. See the history of this page for a list of all contributions to it.