Torus front

Definition 20.15. Consider the action of the group hxT, on the plane, where the standard generators ofhxh act hy vector translations, with vectors ( — 1,1) and m,n). An (m, n)-torus front (or sometimes simply torus front is an orbit of this (Z x h)-action on the set of all (m, n)-grid paths. 

The reason for choosing the word torus here is that the quotient space of the plane by this (Z x Z)-action can be viewed as a torus, where each (m, n)-grid path yields a loop following the grid in the northeasterly direction see Figure 20.4. We notice that every torus front has a unique representative starting from the point (0,0). It will soon become clear why we choose to consider the orbits of the action rather than merely considering these representatives of the orbits. 

A norUiwestem sharp corner of a torus front is a vertex that is entered by a northboimd edge and exited by an eastbound edge in the same way,... 

An elementary flip of a torus front is either a replacement of a northwestern sharp corner at (a, h) by a southeastern sharp corner at (a + 1,6 — 1) or vice versa. It may help to think of a torus front as a sort of flexible snake, where the sharp corners (meaning the vertex in the sharp corner, together with the two adjacent edges) can be flipped about (as if using ball-and-socket joints) the diagonal line connecting the neighbors of the sharp corner vertex. 

Notice that for = 1 there are no conditions, so TFm,n,i simply has all (m, n)-torus fronts as vertices and all flips as higher-dimensional cubes. It is also important to remark that the term horizontal legs includes the legs of length 1 in other words, for > 1, it is prohibited that the torus front contain two consecutive northbound edges. 

If the flips are done on torus fronts instead of grid paths, then we do not need to consider different special cases, which is the main reason why we chose to replace the paths by the orbits of the (Z x Z)-action. ... 

The width of the torus front T, denoted by w(T), is the distance between the millstones measured along the line x = —y. Clearly, the width cannot be less than /2/2, and the minimum is achieved only when m = n by the torus front, where northbound and eastbound edges alternate at every step. 

As mentioned above, each sharp corner defines an elementary flip of T, during which it remains a sharp corner but changes orientation. If this sharp corner were extreme, then the new sharp corner would be extreme (on the opposite side) if and only if w(T) < /2. Clearly, for any torus front, the elementary flips of northwestern extreme corners are mutually noninterfering, and therefore form one flip the same is true for the southeastern extreme corners. Furthermore, if w T) > V2/2, then these two sets of elementary flips do not interfere with each other either. Indeed, if they did, it would mean that... 

Assume now that w(T) > and consider the total flip of all extreme corners. It is geometrically clear that performing any combination of these elementary flips will not increase the width of the torus front, and that performing all of the northwestern flips, or all of the southeastern flips, simultaneously will for sure decrease it. 

It is easy to construct a very thin torus front for fixed m and n simply start from the point (0,0) eastward and then approximate the line connecting (0,0) with (m,n) as closely as possible, staying on one side of this line (touching the line is permitted). It is clear that if the approximation is not the closest possible, then w T) > a/2. 

It is also important to notice that since we assumed m g — 1) < n, each horizontal leg in a thin torus front has length at least g] this can be proved with the same argument as the one we used to show that flipping all extreme corners preserves that property as well. 

The very thin torus front which we just constructed will have (0,0) as a northwestern extreme corner, and after that the northwestern extreme corners will repeat with the period (m + n)/ gcd(m,n). From this we see that in general, to determine a very thin torus front we just need to specify where on the part of the torus front coimecting two subsequent northwestern extreme corners the origin will he. Therefore, we conclude that there are precisely m + n)/gcd(m, n) very thin torus fronts, and that the width of any very thin (m, n)-torus front depends only on m and n. 

We now extend the definition of width. For any flip F, we define its width w F) to be the maximum of widths of all the torus fronts that are members of F. The concepts of millstones and extreme corners can be extended to flips as well see Figure 20.5. 

Let us now finish our study of torus fronts by analyzing the subcomplex... 

On the other hand, if F is a maximal cell, we have w F) = /2. If we now choose a torus front T belonging to this cell, we will also have w(T) = /2, except for two cases if in each elementary flip we either always choose the northwestern path, or if we always choose the southeastern path. We see that each maximal cell contains exactly two torus fronts of width strictly smaller than a/2, and that these are opposite corners of the cube. Since ThinTO,n,s is connected, the conclusion follows. ... 

The argument using torus fronts with which we computed H ( t,m,3 Z2) consisted in presenting a sequence of collapses leading from TF( j-t,3 to Thint m t,3- Due to isomorphism of cochain complexes over Z2, stated in Proposition 20.14, these collapses could have been performed directly on (A, di). 

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