Pull-push in Godin's HCFT for string topology

Question

I am reading Veronique Godin's famous article "Higher string topology operations" (http://arxiv.org/abs/0711.4859) that demonstrates that the string topology operations on $(H_\bullet(L X), H_\bullet(P X))$ for $X$ a compact oriented smooth manifold are part of a homological 2d TQFT with target space $X$.

The idea of the article is clear and simple, but the realization is quite technical. I have the following question on a technical detail.

The idea is that for $\Gamma$ some suitable surface with incoming and outgoing boundary inclusions

$$ \partial_{in} \Gamma \stackrel{in}{\to} \Gamma \stackrel{out}{\leftarrow} \partial_{out} \Gamma $$

we form the mapping space correspondence

$$ X^{\partial_{in} \Gamma} \stackrel{X^{in}}{\leftarrow} X^\Gamma \stackrel{X^{out}}{\to} X^{\partial_{out} \Gamma} $$

Then the string operations and their generalizations are supposed to be given by pull-push of homology classes through this span

$$ out_* \circ in^! : H_\bullet(X^{\partial_{in} \Gamma}) \to H_{\bullet+\chi\Sigma-dim X}(X^{\partial_{out} \Gamma}) \,, $$

where the wrong-way map $in^!$ is "dual fiber integration" via Thom collaps and Thom isomorphism. Or rather, the idea is to have these operations be parameterized by the moduli space of surfaces.

While nice and simple, this idea is maybe a bit too simple: it is hard to get the wrong-way map on homology for $X^{\Gamma} \to X^{\partial_{in} \Gamma}$ under control. Concretely, we'd need to fatten it to an embedding and then find a tubular neighbourhood, both of which is subtle for these infinite-dimensional mapping spaces.

So instead, in Godin's article on p. 22, around diagram (6), this morphism is replaced itself by a zig-zag. The idea is to first fatten $X^{in}$ by remembering where the extra edges $eE$ and extra vertices $eV$ land, separately, that are not in the incoming boundary, then contract these edges to points, and then finally embed the spaces of these extra vertical and contracted edges into a contractible mapping space $W^{eV \coprod eE}$ induced by an embedding of the manifold $X$ into a vector space $W$.

In total then this produces a long zig-zag replacing the above simple span, that in the labelling used in the article reads

$$ X^{\partial_{in} \Gamma} \times W^{eE \coprod eV} \leftarrow X^{\partial_{in} \Gamma} \times X^{eE \coprod eV} \to X^{\partial_{in} \Gamma} \times P X^{eE} \times X^{eV} \stackrel{X^{in} \times \cdots }{\leftarrow} X^{X^\Gamma} \stackrel{X^{out}}{\to} X^{\partial_{out} \Gamma} \,. $$

We can now pull-push-pull-push homology through this longer zig-zag. The contractible factor $W^{eE \coprod eV}$ on the far left doesn't disturb the form of this (if one does it right) and so this produces some map.

I think I understand Godin's article, all the constructions involved in this and the result. Though it is a bit of a tour-de-force, due to some technicalities, not the least of which do arise precisely because of all the extra structure to be taken care of in this longer zig-zag.

So my question finally is: while I see that we can get pull-push through the longer zig-zag to work, how do we know that this longer zig-zag is a good replacement for the naive short and simple zig-zag? Is this justified "only" (not that I doubt that this is a big achievement) by the fact that it works, produces and HQFT and reproduces the string topology operations?

What if I came up with a different long zig-zag, that also makes all these things work? Will it necessarily give the same result?

What if we tried to lift the pull-push construction to the chain level by a chain-level version of the Thom isomorphism, producing a genuine "TCFT" instead of just an "HCFT". Wouldn't that make us want to fall back to the simple one-step pull-push?

Answer 1

Urs, although I answered some of these questions privately, I will post them here as well.

At the moment there is no technology to do the naive short zig-zag you suggest. Godin's techniques for umkehr maps of mapping spaces of maps into a manifold $M$ can do essentially two things: (1) create isolated points in the domain, (2) glue some points in the domain together. The first is done by adding some contractible padding in the form of the Euclidean space $W$ to the domain of the wrong-way arrow and embedding the maps from the new points into $M$ into this Euclidean space; it is then in the correct form to do the standard Thom collapse construction of the umkehr map. The second uses propagating flows, which have as a necessary condition that the map between mapping spaces is giving by the pullback of a finite-dimensional embedding, such that heuristically is of finite codimension. If one tries to identify for examples intervals, this is no longer the case. In Godin's zig-zag, the first wrong-way arrow uses (1), the second uses (2).

We don't know how to do your short zig-zag yet, because if the complement of the incoming boundary graph contains a loop, then we cannot create it directly by creating a point and expanding it. One would get \begin{equation}M^{\partial_{in} \Gamma} \times W \leftarrow M^{\partial_{in} \Gamma \sqcup *} \leftarrow M^{(\partial_{in} \Gamma) \sqcup (\Gamma \backslash \partial_{in} \Gamma)}\end{equation}

There is no known method to do the umkehr map for the second wrong-way arrow, because it is not a homotopy equivalence and hence we cannot produce an inverse up to homotopy. What Godin does is break the complement into a disjoint union of points and lies graphs, each of which is contractible and can be created by method (1). (Coincidentally, this particular way of breaking up the graph is natural and a good choice to get a grip on the local system appears due to the Thom isomorphism for unoriented virtual bundles.) So Godin's construction seems to me be the minimal one with the current technology that does give HCFT operations which coincide with all previously known string topology operations. This could be taken as justification.

However, it is legitimate to ask the questions: Should this construction work? Why this zig-zag? I think the answer lies in the direction that umkehr maps should be functorial: if $f^!$, $g^!$ and $(g \circ f)^!$ make sense then they should coincide. In the case of Godin's zig-zag, the umkehr maps that appear are in the following order:

\begin{equation}(f_{glue})^! \circ (f_{expand})^* \circ (f_{create})^!\end{equation}

where $f_{create}: M^{\partial_{in} \Gamma} \times M^{eV \sqcup eE} \to M^{\partial_{in} \Gamma} \times W^{eV \sqcup eE}$ creates the additional points, $f_{expand}: M^{\partial_{in} \Gamma} \times M^{eV \sqcup eE} \to M^{\partial_{in} \Gamma} \times M^{eV} \times PM^{eE}$ expands those corresponding to edges into intervals, $f_{glue}:M^\Gamma \to M^{\partial_{in} \Gamma} \times M^{eV} \times PM^{eE}$ glues the new vertices and edges together and to the incoming boundary. Now, $f_{expand}$ is a homotopy equivalence, so we are really looking at a composition of two umkehr maps $(f_{glue})^!$ and $(f_{create})^!$, which together do the same thing the umkehr map for your short zig-zag should do. If the construction of that umkehr map is functorial (as it should be), then they should coincide. This is why I think the construction based on Godin's longer zig-zag gives the same answer as a conjectural simpler zig-zag construction.

For the same reason, any other zig-zag should be the give the same operations as the conjectural construction for a simpler zig-zag and hence be equal to Godin's construction. Of course, this is not a proof. If you give me a different zig-zag, I can probably find a zig-zag of zig-zag's between that and Godin's, which shows that the corresponding operations are equal.