I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$.

Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an embedded framed disk $D^k\subset \mathbb{R}^{n+1}, k < n$, with $D^k \cap X = \partial D^k$. Then I think I can do "ambient" surgery on $D^k$ and get a new manifold $Y^n \subset \mathbb{R}^{n+1}$ that is topologically the result of k-surgery on $X$ using the induced framing on $\partial D^k \subset X$ from the framing on $D^k \subset \mathbb{R}^{n+1}$.

Any $D^k \le n/2$ can be embedded into $\mathbb{R}^{n+1}$ by Whitney's trick. We can start with $B^{n+1} \subset \mathbb{R}^{n+1}$ and attach handles of index $k \le n/2$ and get some new manifold $W$ and its boundary $\partial W$ is closed and codimension $1$. So it seems the homotopy type of any $n/2$-CW complex can be realized by an $(n+1)$-manifold embedded into $\mathbb{R}^{n+1}$. However, it is not true that any $(n+1)$-manifold (with boundary) of homotopy type a $n/2$-CW complex can be embedded into $\mathbb{R}^{n+1}$. For example, Stiefel-Whitney classes, which come from the framing of handles, are obstructions; so for our $(n+1)$-manifolds that embed into $\mathbb{R}^{n+1}$, it seems we are not free to pick the framing. Also, intersection form is another obstruction. In general, what are some other examples of manifolds with given homotopy type that can be embedded?

For example, we know that for $n \ge 4$, all finitely-generated groups $G$ can be realized as the fundamental group of some n-manifold. Is this true with the additional requirement that the manifold is a hypersurface of $\mathbb{R}^{n+1}$? I think this is true and the manifold can be taken to be the boundary of an $(n+1)$-manifold only with handles of index $1,2,3$.


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