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Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism $$(ev,\Phi):\mathcal{M}_{0,1}(X,\beta) \to X \times \mathcal{M}_{0,1}$$ converge to the homotopy type of the double loop space of $X$.

Question. So what about higher genus?

I believe that Segal original paper The topology of spaces of rational functions dealt with a genus $g$ Riemann surface and not neccesarily just the genus zero cases. But most of the later papers either have "interesting" target spaces $X \neq \mathbb{CP}^n$ or higher genus - but not both (at least those I was able to find...)

1. What does the Cohen-Jones-Segal conjecture implies for the homotopy type of fibers $$(ev,\Phi):\mathcal{M}_{g,1}(X,\beta) \to X \times \mathcal{M}_{g}$$ with $g \geq 1$? I assume it should be something like: "the inclusion into the space of differentiable maps from a genus $g$ surface and homology class $\beta$ to $X$ is a weak homotopy equivalence"?

2. Am I correct in assuming that such a conjecture should imply that the only sources of $\pi_1$ in the fibers should come from universal Abel map to the relative Picard variety?

Even more vaguely:

1. Does it make sense to try and use Costello's paper Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products to deduce such a statement from a corresponding genus zero Cohen-Jones-Segal for the product stack $S^{g+1}(X)$?

Is there an example of work in this direction?

2. Are there any results coming from h-principle/oka theory (i.e., results about the inclusion into the space of differentiable maps $Hol(S_g,N) \to Map(S_g,N)$ being weak homotopy equivalence for certain Stein manifolds $N$ etc), which can be extended to the case of a projective target space?

( This is basically Part B of my previous question: What is the fundamental group of Kontsevich's space of stable maps? following Jason's extremely helpful suggestions in the comments )

Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism $$(ev,\Phi):\mathcal{M}_{0,1}(X,\beta) \to X \times \mathcal{M}_{0,1}$$ converge to the homotopy type of the double loop space of $X$.

Question. So what about higher genus?

I believe that Segal original paper The topology of spaces of rational functions dealt with a genus $g$ Riemann surface and not neccesarily just the genus zero cases. But most of the later papers either have "interesting" target spaces $X \neq \mathbb{CP}^n$ or higher genus - but not both (at least those I was able to find...)

1. What does the Cohen-Jones-Segal conjecture implies for the homotopy type of fibers $$(ev,\Phi):\mathcal{M}_{g,1}(X,\beta) \to X \times \mathcal{M}_{g}$$ with $g \geq 1$? I assume it should be something like: "the inclusion into the space of differentiable maps from a genus $g$ surface and homology class $\beta$ to $X$ is a weak homotopy equivalence"?

2. Am I correct in assuming that such a conjecture should imply that the only sources of $\pi_1$ in the fibers should come from universal Abel map to the relative Picard variety?

Even more vaguely:

1. Does it make sense to try and use Costello's paper Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products to deduce such a statement from a corresponding genus zero Cohen-Jones-Segal for the product stack $S^{g+1}(X)$?

Is there an example of work in this direction?

2. Are there any results coming from h-principle/Oka theory (i.e., results about the inclusion into the space of differentiable maps $Hol(S_g,N) \to Map(S_g,N)$ being weak homotopy equivalence for certain Stein manifolds $N$ etc), which can be extended to the case of a projective target space?

( This is basically Part B of my previous question: What is the fundamental group of Kontsevich's space of stable maps? following Jason's extremely helpful suggestions in the comments )

Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of $ev:\mathcal{M}_{0,1}(X,\beta) \to X \times \mathcal{M}_{0,1}$ converge to the homotopy type of the double loop space of $X$. So what about higher genus?

I believe that Segal original paper The topology of spaces of rational functions dealt with a genus $g$ Riemann surface and not neccesarily just the genus zero cases. But most of the later papers (with "interesting" target spaces $X \neq \mathbb{CP}^n$) I could find seem to focus on holomorphic spheres...

1. What does the Cohen-Jones-Segal conjecture implies for the homotopy type of fibers $ev:\mathcal{M}_{g,1}(X,\beta) \to X \times \mathcal{M}_{g}$ with $g \geq 1$? (I assume it should be something like: the inclusion into the space of differentiable maps from a genus $g$ surface and homology class $\beta$ to $X$ is a weak homotopy equivalence? ...)

2. Am I correct in assuming that such a conjecture should imply that the only sources of $\pi_1$ in the fibers should come from universal Abel map to the relative Picard variety?

Even more vaguely:

1. Does it make sense to try and use Costello's paper Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products to deduce such a statement from a corresponding genus zero Cohen-Jones-Segal for the product stack $S^{g+1}(X)$?

Is there an example of work in this direction?

2. Are there any results coming from h-principle/oka theory (i.e., results about the inclusion into the space of differentiable maps $Hol(S_g,N) \to Map(S_g,N)$ being weak homotopy equivalence for certain Stein manifolds $N$ etc), which can be extended to the case of a projective target space?

( This is basically Part B of my previous question: What is the fundamental group of Kontsevich's space of stable maps? following Jason's extremely helpful suggestions in the comments )

Let $X$ be a projective variety. As I've been told, there is a conjecture (by Cohen-Jones-Segal) which implies that the homotopy type of fibers of the stablization-evaluation morphism $$(ev,\Phi):\mathcal{M}_{0,1}(X,\beta) \to X \times \mathcal{M}_{0,1}$$ converge to the homotopy type of the double loop space of $X$.

Question. So what about higher genus?

I believe that Segal original paper The topology of spaces of rational functions dealt with a genus $g$ Riemann surface and not neccesarily just the genus zero cases. But most of the later papers either have "interesting" target spaces $X \neq \mathbb{CP}^n$ or higher genus - but not both (at least those I was able to find...)

1. What does the Cohen-Jones-Segal conjecture implies for the homotopy type of fibers $$(ev,\Phi):\mathcal{M}_{g,1}(X,\beta) \to X \times \mathcal{M}_{g}$$ with $g \geq 1$? I assume it should be something like: "the inclusion into the space of differentiable maps from a genus $g$ surface and homology class $\beta$ to $X$ is a weak homotopy equivalence"?

2. Am I correct in assuming that such a conjecture should imply that the only sources of $\pi_1$ in the fibers should come from universal Abel map to the relative Picard variety?

Even more vaguely:

1. Does it make sense to try and use Costello's paper Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products to deduce such a statement from a corresponding genus zero Cohen-Jones-Segal for the product stack $S^{g+1}(X)$?

Is there an example of work in this direction?

2. Are there any results coming from h-principle/oka theory (i.e., results about the inclusion into the space of differentiable maps $Hol(S_g,N) \to Map(S_g,N)$ being weak homotopy equivalence for certain Stein manifolds $N$ etc), which can be extended to the case of a projective target space?

( This is basically Part B of my previous question: What is the fundamental group of Kontsevich's space of stable maps? following Jason's extremely helpful suggestions in the comments )