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I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicated. Since $4=2^2$, this is one of the dimensions where the real projective space $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.

I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicated. Since $4=2^2$, this is one of the dimensions where the real projective space $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.

I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicate. Since $4=2^2$, this is one of the dimensions where real projective plane $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.

I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicated. Since $4=2^2$, this is one of the dimensions where the real projective space $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.

I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicate. Since $4=2^2$, this is one of the dimensions where real projective plane $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fang and Fuquan.

I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.

Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.

It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.

The question in dimension 4 seems more complicate. Since $4=2^2$, this is one of the dimensions where real projective plane $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.