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If $M$ is a connected compact $2$-manifold, then it unknots in $\Bbb R^5$. More generally, $k$-connected $n$-manifolds embed in $\Bbb R^{2n-k}$ and unknot in $\Bbb R^{2n-k+1}$ in the metastable range $m>\frac{3(n+1)}2$. This was proved around 1961 - by Roger Penrose, J.H.C. Whitehead, and Zeeman in the PL category; and by Haefliger in the smooth category. Later Zeeman and Irwin extended the PL result to codimension three (see Zeeman's "Seminar on Combinatorial Topology").

On the other hand, the disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $p\sqcup q:S^2\sqcup S^2\to\Bbb R^5$. A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$. Since $S^2$ unknots in $\Bbb R^5$, the exterior of one component is always homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

By Haefliger's theorem (1963) that embeddings in the metastable range are classified by equivariant homotopy of two-point configuration spaces, the linking number for each pair of components is the only invariant of smoothly embedded $2$-manifolds in $\Bbb R^5$. This recovers the result that connected surfaces unknot in $\Bbb R^5$; and additionally implies that there is nothing new for $3$-component links. [In contrast, there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

Also smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds in the metastable range $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$). In more detail, Haefliger's classification theorem implies that if two smooth embeddings in the metastable range are isotopic (=homotopic through topological embeddings, possibly wild) then they are smoothly isotopic. Weber's PL classification theorem (1967) implies additionally that every PL embedding of a smooth manifold in the metastable range is ambient isotopic to a smooth embedding. Also it follows from results of Edwards and Bryant that an arbitrary topological embedding in codimension $\ge 3$ is isotopic to a PL embedding, and, from results of Bryant-Seebeck, that a locally flat topological embedding is ambient isotopic to a PL embedding.

If $M$ is a connected compact $2$-manifold, then it unknots in $\Bbb R^5$. More generally, $k$-connected $n$-manifolds embed in $\Bbb R^{2n-k}$, provided $k<\frac{n-2}2$, and unknot in $\Bbb R^{2n-k+1}$, provided $k<\frac{n-1}2$. This was proved around 1961 - by Roger Penrose, J.H.C. Whitehead, and Zeeman in the PL category; and by Haefliger in the smooth category. Later Zeeman and Irwin relaxed the metastable dimension restrictions in the PL result to codimension $\ge 3$ (see Zeeman's "Seminar on Combinatorial Topology").

On the other hand, the disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $f\sqcup g:S^2\sqcup S^2\to\Bbb R^5$. A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$. Since $S^2$ unknots in $\Bbb R^5$, the exterior of one component is always homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

By Haefliger's theorem (1963) that embeddings in the metastable range are classified by equivariant homotopy of two-point configuration spaces, the linking number for each pair of components is the only invariant of smoothly embedded $2$-manifolds in $\Bbb R^5$. This recovers the result that connected surfaces unknot in $\Bbb R^5$; and additionally implies that there is nothing new for $3$-component links. [In contrast, there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

Also smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds in the metastable range $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$). In more detail, Haefliger's classification theorem implies that if two smooth embeddings in the metastable range are isotopic (=homotopic through topological embeddings, possibly wild) then they are smoothly isotopic. Weber's PL classification theorem (1967) implies additionally that every PL embedding of a smooth manifold in the metastable range is ambient isotopic to a smooth embedding. Also it follows from results of Edwards and Bryant that an arbitrary topological embedding in codimension $\ge 3$ is isotopic to a PL embedding, and, from results of Bryant-Seebeck, that a locally flat topological embedding in codimension $\ge 3$ is ambient isotopic to a PL embedding.

The disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is, the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $p\sqcup q:S^2\sqcup S^2\to\Bbb R^5$. I'm afraid the linking number is the only invariant here. A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$.

  Since $S^2$ unknots in $\Bbb R^5$ (see below), the exterior of one component is homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

Now if $M$ is a connected compact $2$-manifold, then it PL unknots in $\Bbb R^5$; more generally, $k$-connected $n$-manifolds PL embed in $\Bbb R^{2n-k}$ and PL unknot in $\Bbb R^{2n-k+1}$, as long as the codimension is at least three, by Penrose-Whitehead-Zeeman and Irwin. Also by Haefliger (1963), smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds, where $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$).

EDIT: More interesting are multi-component links of a bunch of $2$-manifolds in $\Bbb R^5$. Firstly, the isotopy classification reduces to homotopy (=link homotopy) classification. Given a link homotopy, we may view it as a link map of the cylinders $M^2\times I$ into $\Bbb R^5\times I$, and then generically it has at most a finite number of double points. Again all double points can be eliminated by the Penrose-Whitehead-Zeeman trick.

Next, I would think that the homotopy classification reduces to that of $2$-component sublinks. Certainly this is so for $3$-component spherical links, as shown by Massey (Topology Appl. 34 (1990), 269–300). [Note that there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

For multi-component spherical links, Koschorke's $\kappa$-invariant is trivial, as long as all $2$-component sublinks are trivial (Topology 36 (1997), 301-324) and I'm quite sure that in this dimension this should imply that the link is homotopically trivial (hence trivial).

If $M$ is a connected compact $2$-manifold, then it unknots in $\Bbb R^5$. More generally, $k$-connected $n$-manifolds embed in $\Bbb R^{2n-k}$ and unknot in $\Bbb R^{2n-k+1}$ in the metastable range $m>\frac{3(n+1)}2$. This was proved around 1961 - by Roger Penrose, J.H.C. Whitehead, and Zeeman in the PL category; and by Haefliger in the smooth category. Later Zeeman and Irwin extended the PL result to codimension three (see Zeeman's "Seminar on Combinatorial Topology").

On the other hand, the disjoint union of two $2$-spheres is a compact $2$-manifold. It definitely knots in $\Bbb R^5$ as detected by the linking number. That is the degree $\alpha$ of $S^2\times S^2\to S^4$, $(p,q)\mapsto \frac{f(p)-g(q)}{||f(p)-g(q)||}$, calling our link $p\sqcup q:S^2\sqcup S^2\to\Bbb R^5$. A nontrivial link is the Hopf link, whose components are the factors of the join $S^5=S^2*S^2$. Since $S^2$ unknots in $\Bbb R^5$, the exterior of one component is always homotopy equivalent to $S^2$, and the linking number is also the degree $\lambda$ of $p(S^2)\to S^5\setminus q(S^2)\simeq S^2$. [In different dimensions, where $\alpha$ and $\lambda$ are not numbers but homotopy classes of spheroids (more precisely $\alpha$ factors though a spheroid up to homotopy, upon killing the wedge), their relation is more interesting: $\alpha$ equals the suspension of $\lambda$ (up to a sign).]

By Haefliger's theorem (1963) that embeddings in the metastable range are classified by equivariant homotopy of two-point configuration spaces, the linking number for each pair of components is the only invariant of smoothly embedded $2$-manifolds in $\Bbb R^5$. This recovers the result that connected surfaces unknot in $\Bbb R^5$; and additionally implies that there is nothing new for $3$-component links. [In contrast, there are the Borromean rings of three $3$-spheres in $\Bbb R^6$, whose nontriviality is detected e.g. by a nonvanishing triple Massey product in the complement. Thinking of the usual Borromean rings in $\Bbb R^3$ as lying in the three coordinate planes, one can similarly do three copies of $S^1*S^1$ lying in the two-factor subproducts of $\Bbb R^2\times\Bbb R^2\times\Bbb R^2$.]

Also smooth, PL and topological knot theories coincide for smooth $n$-manifolds in smooth $m$-manifolds in the metastable range $m>\frac{3(n+1)}2$ (this includes $2$-manifolds in $\Bbb R^5$). In more detail, Haefliger's classification theorem implies that if two smooth embeddings in the metastable range are isotopic (=homotopic through topological embeddings, possibly wild) then they are smoothly isotopic. Weber's PL classification theorem (1967) implies additionally that every PL embedding of a smooth manifold in the metastable range is ambient isotopic to a smooth embedding. Also it follows from results of Edwards and Bryant that an arbitrary topological embedding in codimension $\ge 3$ is isotopic to a PL embedding, and, from results of Bryant-Seebeck, that a locally flat topological embedding is ambient isotopic to a PL embedding.

EDIT: Here is a proof that connected $2$-manifolds PL unknot in $\Bbb R^5$, after Penrose-Whitehead-Zeeman. This is a bit easier than the Whitney trick. Pick a generic homotopy between two embeddings. Viewed as a map $M^2\times I\to\Bbb R^5$, it has as most finitely many double points. A double point is the image of a pair of points, which can be connected by an arc by the hypothesis. The image of the arc is then an embedded $S^1$, and by general position it bounds a $2$-disk in $\Bbb R^5$ that is disjoint from the image of $M$ except at its boundary circle. A small regular neighborhood of this disk is a $5$-ball, and its preimage in $M$ is a regular neighborhood of the arc, which is a $2$-ball. The boundary of the $2$-ball goes into the boundary of the $5$-ball, and we're free to redefine the map on the interior of the $2$-ball in a conewise fashion - so that the double point disappears (by the price of the conical point, which makes it a non-smooth construction). Removing all double points in this way we obtain an embedding.

EDIT: More interesting are multi-component links of a bunch of $2$-manifolds in $\Bbb R^5$. Firstly, the isotopy classification reduces to homotopy (=link homotopy) classification. Given a link homotopy, we may view it as a link map of the cylinders $M^2\times I$ into $\Bbb R^5\times I$, and then generically it has at most a finite number of double points. Again all double points can be eliminated by the Penrose-Whitehead-Zeeman trick.

EDIT: More interesting are multi-component links of a bunch of $2$-manifolds in $\Bbb R^5$. Firstly, the isotopy classification reduces to homotopy (=link homotopy) classification. Given a link homotopy, we may view it as a link map of the cylinders $M^2\times I$ into $\Bbb R^5\times I$, and then generically it has at most a finite number of double points. Again all double points can be eliminated by the Penrose-Whitehead-Zeeman trick.