Gluing a flow line to itself

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 {\mathbb Z}/2 in order to avoid worrying about orientations.

1. Setup.

The setup is this. Given a circle-valued Morse function f on a closed manifold X and a generic metric g, one can define the Novikov complex of (f,g), with coefficients in the ring of Laurent series ({\mathbb Z}/2)((T)). The simplest way to describe this is as follows: The chain complex is the free ({\mathbb Z})/2)(T) module generated by the critical points. To define the differential, let \Sigma be a level set of f which does not contain any critical points. If p is an index i critical point, then

\partial p = \sum_q \sum_{k=0}^\infty n_k(p,q)T^kq

where the sum is over index i-1 critical points q, and n_k(p,q) denotes the mod 2 count of flow lines from p to q that cross \Sigma exactly k 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 f:X\to S^1 and does not depend on the metric g. One way to do so is to consider a generic one-parameter family \{(f_t,g_t)\}_{t\in[-1,1]} of pairs of circle-valued functions and metrics. For some t, called “bifurcation times”, the pair (f_t,g_t) will fail to be Morse-Smale. You then want to understand how the chain complex changes as you pass through this value of t 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 t=0, the function f_0 is Morse, but for the metric g_0, we have a flow line u_0 from an index i critical point q to itself. If there is a flow line u_+ from an index i+1 critical point p to q at time t=0, then we “know” that u_+ and u_0 can be glued to a flow line from p to q either before or after the bifurcation. The difficult part is that at time t=0 we also have, for each positive integer k, broken configurations consisting of the flow line u_+ from p to q, together with the concatenation of k copies of the flow line u_0 from q 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 k copies of u_0 together with a flow line u_- from q to an index i-1 critical point r.

For definiteness suppose that u_0 crosses \Sigma once. Let \partial_- and \partial_+ 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 A=1 + T + \cdots such that

\partial_+ = A_q \circ \partial_- \circ A_q^{-1},

where A_q denotes the operator which multiplies q by A 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

A = (1+T)^{\pm1}.

I will now sketch a direct proof that A=(1+T)^{\pm1}, which should generalize to Floer theory, and in which we actually see how the gluing works.

2. Warmup: Gluing one copy of u_0.

To warm up, let us first consider the simpler problem of gluing u_+ to u_-. I will do this following the obstruction bundle gluing technique in [GLUING], which will then generalize to glue multiple copies of u_0. [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” u_+ and u_- 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 R and t. Here R is a large positive real number, and t is a small positive or negative real number. To preglue, we translate u_+ up and u_0 down so that the total translation distance is R; and we change the time from 0 to t. We then patch the translated curves together using cutoff functions \beta_+ on u_+ and \beta_0 on u_0, which have derivatives of order 1/R.

Now let \psi_+ be a section of u_+^*TX, and let \psi_0 be a section of u_0^*TX. We can then perturb the preglued curve by pushing it off via the exponential of \beta_+\psi_+ + \beta_0\psi_0. Let F_t denote the equation to be a gradient flow line of (f_t,g_t). We can then write F_t of the perturbed preglued curve in the form

F_t(\beta_+\psi_+ + \beta_0\psi_0) = \beta_+\Theta_+(\psi_+,\psi_0,t) + \beta_0\Theta_0(\psi_+,\psi_0,t)

Here \Theta_+ = D_+\psi_+ + \cdots where D_+ denotes the deformation operator associated to u_+, and the additional terms arise from the patching and do not concern us right now. Likewise \Theta_0 = D_0\psi_0 + \cdots where D_0 is the deformation operator associated to u_0.

As in Section 5 of [GLUING], one can show using the contraction mapping theorem that for R large and t small, there is a unique pair (\psi_+,\psi_0) in appropriate Banach spaces such that \psi_+ is orthogonal to Ker(D_+), \psi_0 is orthogonal to Ker(D_0), \Theta_+ is orthogonal to Im(D_+), and \Theta_0 is orthogonal to Im(D_0).

Now since we are in a generic one-parameter family, D_+ is surjective and D_0 has a one-dimensional cokernel. Thus we have achieved \Theta_+=0, while \Theta_0 lives in a one-dimensional space. If in fact \Theta_0=0, then we have successfully glued. If not, then we can think of \Theta_0 as the obstruction to gluing. Let us denote this gluing obstruction by

o(R,t) \in Coker(D_0).

We can think of o as a section of the “obstruction bundle” over the space of gluing parameters (R,t) whose fiber is Coker(D_0) (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 u_+ and u_-, and pairs (R,t) such that the obstruction o(R,t)=0. And for the purpose of understanding the differentials \partial_+ and \partial_-, we want to know: For fixed small t, how many solutions are there to the equation o(R,t)=0?

The next step is to approximate o(R,t). As in Section 8 of [GLUING], one has an approximation

o(R,t) \approx e^{-\lambda_-R}\eta + tF'(u_0)

Here \lambda_- is the smallest positive eigenvalue of the Hessian of q, \eta is determined by the “leading asymptotic coefficient” of the negative end of u_+, and F'(u_0) denotes the projection to Coker(D_0) of \frac{d}{ds}|_{s=0}F_s(u_0). What I mean by “approximation” is that for fixed small t, the left and right hand side have the same count of zeroes (on the set of sufficiently large R). Note that since we are in a generic one-parameter family, we have F'(u_0)\neq 0. Thus, to make things a little simpler, we can choose an identification Coker(D_0)\simeq {\mathbb R} such that F'(u_0) corresponds to 1. Then we can rewrite the above approximation as

o(R,t) \approx e^{-\lambda_- R}\eta + t

where now \eta is a real number determined by the asymptotics of u_+. This real number is nonzero assuming suitable genericity.

Thus, for fixed small t we want to count solutions R to the equation

e^{-\lambda_-R}\eta + t = 0.

We see immediately that if t has the same sign as \eta, then there is no solution, while if t and \eta have opposite signs then there is one solution.

In conclusion, u_+ and u_0 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 u_+.

3. Gluing two copies of u_0.

Now let’s try to glue u_+ to two copies of u_0. In this situation we have three gluing parameters R_1, R_2, and t. As before, t is a small positive or negative real number by which we shift time. Furthermore, R_1 is the translation distance between u_+ and the upper copy of u_0, while R_2 is the translation distance between the two copies of u_0. We now have to count solutions to the equations

e^{-\lambda_- R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t=0,

e^{-\lambda_- R_2}\eta_- + t=0.

Here \eta and \lambda_- are as before. In addition, \lambda_+ is minus the smallest negative eigenvalue of the Hessian of q, while \eta_+ and \eta_- are determined by the “leading asymptotic coefficients” associated to the positive and negative ends of u_0, respectively. Under suitable genericity assumptions, \lambda_+\neq \lambda_-.

For a fixed small t, there is exactly one pair (R_1,R_2) satisfying these equations, provided that all of the following four conditions hold (and otherwise there is no solution):

(i) \eta_- and t have opposite signs.

(ii) If \eta_+ and t have the same sign, then \eta and t have opposite sign.

(iii) If \eta_+ and t have opposite signs, and if \eta and t have opposite signs, then \lambda_+ > \lambda_-.

(iv) If \eta_+ and t have opposite signs, and if \eta and t have the same sign, then \lambda_- < \lambda_+.

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 u_0.

It’s slightly simpler to glue u_+ to k\ge 3 copies of u_0. Now we have gluing parameters (R_1,\ldots,R_k,t), where R_1 denotes the translation distance between u_+ and the top copy of u_0, while for j=2,\ldots,k, the gluing parameter R_j is the translation distance between the (j-1)^{st} copy of u_0 and the j^{th} copy, counting downward. The (approximate) gluing equations are

e^{-\lambda_-R_1}\eta + e^{-\lambda_+R_2}\eta_+ + t = 0,

e^{-\lambda_-R_j}\eta_- + e^{-\lambda_+R_{j+1}}\eta_+ + t=0 for j=2,\ldots,k-1,

e^{-\lambda_-R_k}\eta_- + t = 0.

For fixed small t, these equations have one solution (R_1,\ldots,R_k) when all of the following three conditions hold, and otherwise they have no solution:

(i) \eta_- and t have opposite signs.

(ii) If \eta_+ and t have opposite signs then \lambda_+>\lambda_-.

(iii) \eta and t 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 u_+, define a power series f_+(u_+) = 1 + \sum_{k=1}^\infty a_kT^k where a_k is the number of gluings of u_+ to k copies of u_0 for t>0. Likewise define f_-(u_+) to be the power series whose k^{th} coefficient is the number of gluings of u_+ to k copies of u_0 for t<0.

In addition, if u_- is a flow line from q to an index i-1 critical point r, define g_+(u_-) and g_-(u_-) to be the power series whose k^{th} coefficient is the number of gluings of k copies of u_0 to u_- for t>0 and t<0 respectively.

If we did this right, then we should have

f_+(u_+) / f_-(u_+) = g_-(u_-) / g_+(u_-) = A

where A does not depend on u_+ or u_- and is the power series we are looking for, which we expect to equal (1+T)^{\pm 1}. Let’s see.

First let’s calculate f_+(u_+) and f_-(u_+). There are various cases depending on the signs of \eta, \eta_+, \eta_-, and \lambda_+-\lambda_-.

Without loss of generality \eta_-<0.

Suppose \eta_+>0. If \eta>0 then f_+=1 (because no gluing is possible for positive time) and f_-=1+T (because gluing for negative time is possible only for k=1). If \eta<0 then f_+=(1+T)^{-1}=1+T+T^2+\cdots (because for positive time gluing is possible for any k) and f_-=1. Either way, we have

f_+(u_+)/f_-(u_+) = (1+T)^{-1}.

If \eta_+<0 and \lambda_+>\lambda_- then f_+ and f_- are the same as in the previous case.

The last and most interesting case is where \eta_+<0 and \lambda_+ < \lambda_-. If \eta>0, then gluing for positive time is only possible for k=2, so we get f_+=1+T^2; while gluing for negative time is possible only for k=1, so f_-=1+T. If \eta<0, then we have f_+=1+T and f_-=1. Either way,

f_+(u_+) / f_-(u_+) = 1+T.

One can compute g_+(u_-) and g_-(u_-) by symmetry arguments and check that g_-/g_+ = f_+/f_-; 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 \Sigma once.


This entry was posted in Uncategorized. Bookmark the permalink.

3 Responses to Gluing a flow line to itself

  1. Paolo says:

    Hi Michael, this is a very interesting post. Could you also explain how one arrives to the approximation of o(R,t), please?

    • When you preglue, you have to patch together the two flow lines using cutoff functions. The failure of the patched flow line to satisfy the flow line equation includes a term which is basically the derivative of \beta_+ times \psi_+. And this is well approximated by the leading term in the asymptotic expansion of \psi_+. That’s the basic idea; the details get rather complicated.

  2. Paolo says:

    Hi Michael, sorry for resuscitating an old post. I’ve been trying to fill in the details of this computation as a warm up for the ECH gluing. The easiest way I found to obtain the equation
    \beta_+ \Theta_+ + \beta_0 \Theta_0 =0 was to take \beta_+ and \beta_0 to be a partition of unity and form the preglued curve as \beta_+ u_+ + \beta_0 u_0. (Here I’m being sloppy and taking coordinates around the image of u_+ and u_0 in order to avoid dealing with exponentials.) The problem with this choice of pregluing is that in \Theta_0 I have a term \beta_+' \psi_+ and a term \beta_0' \psi_0 and I can’t understand why only the first one matters. (And by the way, are you sure you meant \psi_+ and not u_+ in your answer from August 25?)

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s