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Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the each component of the embedding space should be of finite type even without the hypothesis $2m+2\le n$ (but with $m \le n-3$). However, I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n-m) \times_{BG} BO $$ through $BO(n-m)$. Here $BG$ classifies stable spherical fibrations. Rationally, the space can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.

Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the each component of the embedding space should be of finite type even without the hypothesis $2m+2\le n$ (but with $m \le n-3$). However, I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n{-}m) \times_{BG} BO $$ through $BO(n{-}m)$. Here $BG$ classifies stable spherical fibrations, $BG(k)$ classifies $(k-1)$-spherical fibrations, and the displayed target is meant to be a homotopy pullback. Rationally, the space of such lifts can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.

Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the result should still be tree without the hypothesis $2m+2\le n$, but $m \le n-3$ I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n-m) \times_{BG} BO $$ through $BO(n-m)$. Here $BG$ classifies stable spherical fibrations. Rationally, the space can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.

Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the each component of the embedding space should be of finite type even without the hypothesis $2m+2\le n$ (but with $m \le n-3$). However, I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n-m) \times_{BG} BO $$ through $BO(n-m)$. Here $BG$ classifies stable spherical fibrations. Rationally, the space can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.

Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the result should still be tree without the hypothesis $2m+2\le n$, but $m \le n-3$ I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n-m) \times_{BG} BO $$ through $BO(n-m)$. Rationally, this space might be computable.

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.

Here are some comments:

1. Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the result should still be tree without the hypothesis $2m+2\le n$, but $m \le n-3$ I do not know how to prove this more general statement.

2. Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n-m) \times_{BG} BO $$ through $BO(n-m)$. Here $BG$ classifies stable spherical fibrations. Rationally, the space can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3. A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.