Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are denoted $J^rY$ ($r\in\mathbb N$, $J^0Y=Y$), and the infinite jet bundle $J^\infty Y$ is - as a topological space - the projective limit of the system $(J^rY,\pi^r_s)$ where $\pi^r_s:J^rY\rightarrow J^sY$, $r\ge s$ is the standard projection.
In the literature, there exists several ways through which $J^\infty Y$ acquires a smooth manifold-like structure. What is considered a smooth function on $J^\infty Y$ depends heavily on the choice of this smooth structure. I am confused by the wide variety of these definitions and I often have trouble establishing equivalence. First I will list some examples I have encountered.
- In [Saunders], $J^\infty Y$ is defined as a Fréchet manifold built on the vector space $\mathbb R^\infty$ which arises as the projective limit of $\mathbb R^n$s with respect to the system $p^n_{n-1}:\mathbb R^n\rightarrow\mathbb R^{n-1}$, $p^n_{n-1}(x^1,\cdots,x^{n-1},x^n)=(x^1,\cdots,x^{n-1})$. Smooth functions are determined by what smooth functions are on $\mathbb R^\infty$, and their "finite-orderness" is characterized by the fact that such smooth functions have only a finite number of nonzero partial derivatives at each point. I don't know whether this structure admits partitions of unity (Saunders doesn't go into it) but I find it likely that it does.
- In [Takens], no systematic smooth structure is given on $J^\infty Y$, he takes the smooth functions to be those which are locally the pullbacks of smooth functions from a finite order jet bundle, i.e. $f:J^\infty Y\rightarrow\mathbb R$ is smooth if for each $q\in J^\infty Y$ there is an $r$) and an $U^r\in J^rY$ open set with $q\in (\pi^\infty_r)^{-1}(U^r)$ and a smooth function $f_r\in C^\infty(U^r)$ such that $f|_{(\pi^\infty_r)^{-1}(U^r)}=f_r\circ\pi^\infty_r$. Takens then proves that there is a partition of unity subordinate to every open cover of $J^\infty Y$. I also think that this definition allows one to define smooth functions over every $O\subseteq J^\infty Y$ open set as the open sets of $J^\infty Y$ are generated by open sets of $J^rY$ for all $r$.
- In [Anderson], no systematic smooth structure is given either. Instead, he defines smooth functions to be the (globally) finite order ones, i.e. $f:J^\infty Y\rightarrow\mathbb R$ is smooth if and only if there is an $r$ and a smooth function $f_r\in C^\infty(J^rY)$ such that $f=f_r\circ\pi^\infty_r$. Anderson states that if the notion of smooth function was extended to the locally finite order ones then there would be smooth functions that aren't continuous. This definition seems to be the most convenient for calculus of variations, but I have some concerns. For example it seems non-obvious me how to define the space $C^\infty(O)$ where $O$ is an arbitrary open set of $J^\infty Y$. I guess a Takens-like approach can be taken but then it seems the association $O\mapsto C^\infty(O)$ defines only a presheaf as the glueing of finite order functions only need to be locally finite order. Moreover it seems that the association $W\mapsto C^\infty(W^\infty)$ where $W\subseteq Y$ is open and $W^\infty=(\pi^\infty_0)^{-1}(W)$ also defines only a presheaf over $Y$ for the same reason.
- In [Güneysu] pro-finite dimensional manifolds are defined and the infinite jet bundle is given as a pro-finite dimensional manifold.The definition of a smooth function seems to be same as in [Takens], with the added difference that a smooth function is demanded to be also continuous (Takens does not postulate continuity and based on the aforementioned remark in [Anderson] it seems that this added assumption is not superfluous).
And now come the problems:
- There are a couple of remarks on nlab which seems to contradict many of these. For example in https://ncatlab.org/nlab/show/jet+bundle#concrete they say
but one has to decide in which category of infinite-dimensional manifolds to take this limit: 1) one may form the limit formally, i.e. in pro-manifolds. This is what is implicit for instance in Anderson, p.3-5; 2) one may form the limit in Fréchet manifolds, this is farily explicit in (Saunders 89, chapter 7). See at Fréchet manifold – Projective limits of finite-dimensional manifolds. Beware that this is not equivalent to the pro-manifold structure (see the remark here). It makes sense to speak of locally pro-manifolds.
and in the link in this remark states
Beware, that infinite jet bundles are also naturally thought of as pro-manifolds. This differs from the Frechet manifold structure of example 4.5: A morphism of pro-manifolds is equivalently a function that is “globally of finite order”, in that [...] But by prop. 4.4 a morphisms of Fréchet manifolds is only restricted to have finite order of partial derivatives at every point. This is a weaker condition. In fact it seems to be also weaker than the condition of being “locally of finite order” considered in Takens 79. Hence it makes sense to speak of locally pro-manifolds.
So it seems that nlab's definition of a pro-manifold (I prefer saying pro-finite dimensional manifold but I meant what is supposed to be the same thing) is different from that of [Güneysu], on the other hand the pro-manifold page on nlab has no references, so I don't know what is precisely the framework they are working in.
- The second problem I have is the paper [GMS]. I will try to briefly summarize its contents. The authors seem to be comparing the cohomology of the variational bicomplex in the Takens-like approach (differential forms are locally finite order) and the Anderson-like approach (differential forms are globally finite order). They make the claim that so far only the locally finite order variational bicomplex (LFOVB) had its cohomology calculated and most approaches that work in this case do not work for the globally finite order variational bicomplex (GFOVB). They then proceed to prove that the GFOVB has the same cohomology as the LFOVB by relating the cohomology of the former to the latter. $$ \ $$ Their argument seems to be around the fact that most cohomology calculations (eg. in [Takens] but also in the paper by Anderson and Duchamp which predates [Anderson] and also in [Krupka] for finite order jet bundles) are taken by considering a fine sheaf of differential forms that are defined on a jet bundle but the sheaf is over $Y$, i.e. a sequence of the form $$ 0\longrightarrow\mathbb R_Y\longrightarrow \mathcal O^0\longrightarrow\cdots\longrightarrow\mathcal O^r\longrightarrow\cdots, $$ where each $\mathcal O^r$ is a sheaf over $Y$ whose sections are differential forms over $J^\infty Y$ or $J^sY$ (for some $s$) and using sheaf cohomology together with the abstract de Rham theorem. However the globally finite order forms form a separated presheaf whose sheafification is the sheaf of locally finite order forms (I think?) and they don't have partitions of unity which throws a wrench in this argument for the GFOVB. $$ \ $$ However [GMS] postdates [Anderson] by quite a few years and Anderson does calculate the cohomology of the GFOVB in [Anderson]. He uses techniques other than the aforementioned sheafy approach though. [GMS] never refers to [Anderson] though (they do refer to Anderson/Duchamp). $$ \ $$ In light of the above I question how correct the global calculations are in [Anderson]. Anderson only uses paritions of unity on $Y$ but upon thinking about it I get the feeling his various glueing arguments are faulty as they might result in locally finite order forms rather than globally finite order ones. At any rate it seems to me that [Anderson] and [GMS] contradict one another thus cannot be both correct (unless of course I am missing something).
Questions:
I'd like to make some sense into this mess so lets see.
- How many inequivalent definitions of $J^\infty Y$ as a manifold-like structure exists? Based on the nlab remarks at least the Saunders/Takens/Anderson approaches seem to be inequivalent to one another.
- Are there multiple inequivalent definitions of pro-finite dimensional manifolds in use? The nlab comments seem to imply that whatever definition they are working with contains the Anderson approach to $J^\infty$ as a subcase but the Güneysu approach seems to be different and closer (but not necessarily equivalent) to the Takens approach.
- Although this is subjective, what structure on $J^\infty Y$ is the best suited for calculus of variations/the variational bicomplex? Basically, I just want to properly formalize the variational bicomplex for myself in a way that it includes only finite-order differential forms (I don't care about Lagrangians with unbounded order) but I got really confused by all these stuff. It also seems to me the Anderson approach might not actually be the best even if it only contains finite order forms if convenient arguments cannot be used. I am basically looking for the most hassle-free and lazy approach possible.
- Assuming Anderson wasn't being sloppy with the definitions in [Anderson], what is the generalization of finite dimensional manifolds into which Anderson's definition of $J^\infty Y$ fits into? (eg. what nlab calls pro-manifolds. But since this terminology seems to be ambigous I cannot just rely on the name alone) I'd like to read more on that structure, even outside jet bundles.
- Regarding the [Anderson]-[GMS] conflict, are the global cohomology calculations in [Anderson] correct? What these to publications say seem to be in conflict and I cannot decide which one is correct. Are they even in conflict actually?
References:
- [Saunders]: D. Saunders - The geometry of jet bundles
- [Takens]: F. Takens - A global version of the inverse problem to the calculus of variations
- [Anderson]: I. M. Anderson - The Variational Bicomplex
- [Güneysu]: B. Güneysu, M. J. Pflaum - The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
- [Krupka]: D. Krupka - Introduction to Global Variational Geometry
- [GMS]: Giachetta, Mangiarotti, Sardanashvily - Cohomology of the variational bicomplex on the infinite order jet space
