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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

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

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?

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?

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

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

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 the 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?

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

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

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?