References for this post:
[TORSION] “Reidemeister torsion in generalized Morse theory”
[LEE] Yi-Jen Lee, “Reidemeister torsion in Floer-Novikov theory and counting pseudoholomorphic tori I, II”
[GLUING] With Cliff Taubes, “Gluing pseudoholomorphic curves along branched covered cylinders II”
In this post, inspired by a question of Jiayong Li and Katrin Wehrheim (and other people have asked me about this earlier), I want to discuss (modulo most analytic details) what happens when you glue multiple copies of a flow line in circle-valued Morse theory to itself. This is a nice example of the obstruction bundle gluing technique which I learned from Cliff Taubes during our work on gluing along branched covers of trivial cylinders [GLUING]. It’s maybe a bit too simple an example since the obstruction bundle in this case is trivial, but it’s fun. (Did I just say that gluing is fun? I must be going nuts.) A similar story holds for holomorphic cylinders, but we will stick with Morse theory to keep things as simple as possible. Also, we will work over in order to avoid worrying about orientations.
The setup is this. Given a circle-valued Morse function on a closed manifold and a generic metric , one can define the Novikov complex of , with coefficients in the ring of Laurent series . The simplest way to describe this is as follows: The chain complex is the free module generated by the critical points. To define the differential, let be a level set of which does not contain any critical points. If is an index critical point, then
where the sum is over index critical points , and denotes the mod 2 count of flow lines from to that cross exactly times. (There are better ways to say this, but I want to focus on gluing and not get confused about algebra at the same time.)
Suppose you want to prove that the Novikov homology depends only on the homotopy class of the map and does not depend on the metric . One way to do so is to consider a generic one-parameter family of pairs of circle-valued functions and metrics. For some , called “bifurcation times”, the pair will fail to be Morse-Smale. You then want to understand how the chain complex changes as you pass through this value of and see that the homology is invariant. I worked out this bifurcation theory long ago in [TORSION] (for other purposes involving Reidemeister torsion), sometimes “cheating” by using some finite dimensional techniques which make things a lot easier. Yi-Jen Lee subsequently extended this to Hamiltonian Floer theory in her papers [LEE].
However there is one bifurcation which I only analyzed indirectly using a “finite cylic cover trick”, and I believe that Yi-Jen used more or less the same trick. The goal of this post is to use obstruction bundle gluing to analyze more explicitly what is going on in this bifurcation.
The bifurcation in question is where at some time, say , the function is Morse, but for the metric , we have a flow line from an index critical point to itself. If there is a flow line from an index critical point to at time , then we “know” that and can be glued to a flow line from to either before or after the bifurcation. The difficult part is that at time we also have, for each positive integer , broken configurations consisting of the flow line from to , together with the concatenation of copies of the flow line from to itself. The question is, how many ways can we glue these to flow lines before and after the bifurcation? Same question for the concatenation of copies of together with a flow line from to an index critical point .
For definiteness suppose that crosses once. Let and denote the differentials before and after the bifurcation. (One has to be a bit careful to make sense of this since the set of bifurcation times is not necessarily discrete. See Section 2.1 of [TORSION].) Lemma 3.7 in [TORSION] shows that there is a power series such that
where denotes the operator which multiplies by and fixes all other critical points. This is all we need to know for most purposes, such as showing that Novikov homology is invariant, or proving invariance of the product of Reidemeister torsion and the zeta function as in [TORSION].
However maybe we want to know what this power series actually is. An extremely obscure reference in Remark 3.12 of [TORSION] implies that
I will now sketch a direct proof that , which should generalize to Floer theory, and in which we actually see how the gluing works.
2. Warmup: Gluing one copy of .
To warm up, let us first consider the simpler problem of gluing to . I will do this following the obstruction bundle gluing technique in [GLUING], which will then generalize to glue multiple copies of . [GLUING] is not exactly light reading, and one could probably write a more readable account of the present simpler situation. However I am just going to explain the basic idea without going into analytic details such as which Banach space completions to choose, how to bound the error terms, etc.
As usual, we will first “preglue” and and then try to perturb the preglued curve to an actual flow line. To preglue, there are two “gluing parameters”, which we can denote by and . Here is a large positive real number, and is a small positive or negative real number. To preglue, we translate up and down so that the total translation distance is ; and we change the time from to . We then patch the translated curves together using cutoff functions on and on , which have derivatives of order .
Now let be a section of , and let be a section of . We can then perturb the preglued curve by pushing it off via the exponential of . Let denote the equation to be a gradient flow line of . We can then write of the perturbed preglued curve in the form
Here where denotes the deformation operator associated to , and the additional terms arise from the patching and do not concern us right now. Likewise where is the deformation operator associated to .
As in Section 5 of [GLUING], one can show using the contraction mapping theorem that for large and small, there is a unique pair in appropriate Banach spaces such that is orthogonal to , is orthogonal to , is orthogonal to , and is orthogonal to .
Now since we are in a generic one-parameter family, is surjective and has a one-dimensional cokernel. Thus we have achieved , while lives in a one-dimensional space. If in fact , then we have successfully glued. If not, then we can think of as the obstruction to gluing. Let us denote this gluing obstruction by
We can think of as a section of the “obstruction bundle” over the space of gluing parameters whose fiber is (it is a trivial bundle in this case).
As in Section 7 of [GLUING], one can show that there is a bijection between gluings of and , and pairs such that the obstruction . And for the purpose of understanding the differentials and , we want to know: For fixed small , how many solutions are there to the equation ?
The next step is to approximate . As in Section 8 of [GLUING], one has an approximation
Here is the smallest positive eigenvalue of the Hessian of , is determined by the “leading asymptotic coefficient” of the negative end of , and denotes the projection to of . What I mean by “approximation” is that for fixed small , the left and right hand side have the same count of zeroes (on the set of sufficiently large ). Note that since we are in a generic one-parameter family, we have . Thus, to make things a little simpler, we can choose an identification such that corresponds to . Then we can rewrite the above approximation as
where now is a real number determined by the asymptotics of . This real number is nonzero assuming suitable genericity.
Thus, for fixed small we want to count solutions to the equation
We see immediately that if has the same sign as , then there is no solution, while if and have opposite signs then there is one solution.
In conclusion, and glue to one flow line. Whether this glued flow line exists before or after the bifurcation depends on the asymptotics of the negative end of .
3. Gluing two copies of .
Now let’s try to glue to two copies of . In this situation we have three gluing parameters , , and . As before, is a small positive or negative real number by which we shift time. Furthermore, is the translation distance between and the upper copy of , while is the translation distance between the two copies of . We now have to count solutions to the equations
Here and are as before. In addition, is minus the smallest negative eigenvalue of the Hessian of , while and are determined by the “leading asymptotic coefficients” associated to the positive and negative ends of , respectively. Under suitable genericity assumptions, .
For a fixed small , there is exactly one pair satisfying these equations, provided that all of the following four conditions hold (and otherwise there is no solution):
(i) and have opposite signs.
(ii) If and have the same sign, then and have opposite sign.
(iii) If and have opposite signs, and if and have opposite signs, then .
(iv) If and have opposite signs, and if and have the same sign, then .
Here (i) is necessary and sufficient to solve the second equation; and then assuming this, (ii)—(iv) are necessary and sufficient to solve the first equation.
In summary, sometime you can glue for positive or negative time, depending on a handful of signs.
4. Gluing three or more copies of .
It’s slightly simpler to glue to copies of . Now we have gluing parameters , where denotes the translation distance between and the top copy of , while for , the gluing parameter is the translation distance between the copy of and the copy, counting downward. The (approximate) gluing equations are
For fixed small , these equations have one solution when all of the following three conditions hold, and otherwise they have no solution:
(i) and have opposite signs.
(ii) If and have opposite signs then .
(iii) and have opposite signs.
Here (i) is needed to solve the last equation, then (ii) is needed to solve the middle equations, and then (iii) is needed to solve the first equation.
5. Putting it all together.
Let us now count and organize what is glued.
Given , define a power series where is the number of gluings of to copies of for . Likewise define to be the power series whose coefficient is the number of gluings of to copies of for .
In addition, if is a flow line from to an index critical point , define and to be the power series whose coefficient is the number of gluings of copies of to for and respectively.
If we did this right, then we should have
where does not depend on or and is the power series we are looking for, which we expect to equal . Let’s see.
First let’s calculate and . There are various cases depending on the signs of , , , and .
Without loss of generality .
Suppose . If then (because no gluing is possible for positive time) and (because gluing for negative time is possible only for ). If then (because for positive time gluing is possible for any ) and . Either way, we have
If and then and are the same as in the previous case.
The last and most interesting case is where and . If , then gluing for positive time is only possible for , so we get ; while gluing for negative time is possible only for , so . If , then we have and Either way,
One can compute and by symmetry arguments and check that ; I will leave this as an exercise.
6. Bonus: closed orbits.
I also leave it as an exercise to similarly check that one (mod 2) simple closed orbit of the gradient flow is created or destroyed by this bifurcation, and this orbit intersects once.