Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time:

1. "Quelques propriétés des variétés-bords", Colloque de Topologie de Strasbourg, 1951, no. V. La Bibliothèque Nationale et Universitaire de Strasbourg, 1952.

2. "Sur les variétés cobordantes", Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 7. La Bibliothèque Nationale et Universitaire de Strasbourg, 1953.

These two notes prefigure some of the results and some of the techniques that will appear slightly later in his 1954 paper "Quelques propriétés globales des variétés différentiables" and each of these notes contains a proof of the fact that $\Omega_3=0$. In fact, these two prior proofs are different from the demonstration that follows by specializing the results of his '54 paper, and which was alluded to in the above answers.

His '52 proof (in the 1953 proceedings) is close to Pontryagin's ideas relating framed cobordism groups to stable homotopy groups of spheres. Thom considers there the map $\psi_k:\pi_{n+k}(S^n) \to \Omega_k$ which consists in taking regular preimages of smooth approximations of maps $S^{n+k} \to S^n$. This map is additive and its image consists of the cobordism classes of those $k$-manifolds which can be embedded in $\mathbb{R}^{n+k}$ with trivial normal bundle. Since any closed orientable $3$-manifold is parallelizable, the map $\psi_3$ is surjective (for $n$ large enough) and it suffices to prove that $\psi_3$ vanishes on a generating set of the group $\pi_{n+3}(S^n)$. To conclude his argument, Thom mentions that this stable homotopy group is isomorphic to $\mathbb{Z}_{24}$ (which, I guess, he knew from Cartan or Serre) and, most importantly, that it is generated by the suspension of the quaternionic Hopf fibration with fiber $S^3= \partial D^4$.

In contrast with his '52 and '54 proofs, his '51 proof (in the 1952 proceedings) does not use any kind of Pontryagin-Thom constructions but, as Lickorish and Rourke will do some years later, he uses the fact that any closed oriented $3$-manifold can be presented by a Heegaard splitting. Let $H_g$ be the handlebody of genus $g$, let $\hbox{MCG}(\partial H_g)$ denote the mapping class group of its boundary and for any $f \in \hbox{MCG}(\partial H_g)$, set

$$ V_f := H_g \cup_f (-H_g). $$

Thom starts by observing that, if $V_f$ and $V_{f'}$ bound, then so does $V_{f \circ f'}$ (see Bruno's comment to Daniel's answer). Next, he recalls that Dehn found an explicit and finite system of generators $\{\tau_i\}$ for $\hbox{MCG}(\partial H_g)$ which consists of several Dehn twists: there are 2, 5 and 2g(g-1) such generators for $g=1$, $g=2$ and $g>2$ respectively. Thus, by the previous observation, it suffices to check that $V_{\tau_i}$ is a boundary for any $g\geq 1$ and for any $i$. For any $g>1$, it can be observed that each of the Dehn twists $\tau_i$ is performed along a curve which avoids a full (solid) handle of $H_g$: hence $V_{\tau_i}$ is the connected sum of $S^1 \times S^2$ with a $3$-manifold admitting a Heegaard splitting of genus $g-1$. Since such a connected sum can be realized in dimension $4$ by the attachement of a handle of index $1$, we are allowed to do an induction on the genus $g$. For $g=1$, the $3$-manifolds $V_{\tau_1}$ and $V_{\tau_2}$ are $S^3$ and $S^1 \times S^2$, which are obviously boundaries.

It seems that this early proof that $\Omega_3=0$ has been forgotten since then: this is probably due to the fact that the proceedings is not widely available and is written in french. Personally, I did not know it before I opened this proceedings in the Math' Library of Strasbourg a few months ago! Nonetheless, Haefliger mentions it in his survey paper of Thom's works (Publ. IHES 1988).

Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time:

1. "Quelques propriétés des variétés-bords", Colloque de Topologie de Strasbourg, 1951, no. V. La Bibliothèque Nationale et Universitaire de Strasbourg, 1952.

2. "Sur les variétés cobordantes", Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 7. La Bibliothèque Nationale et Universitaire de Strasbourg, 1953.

These two notes prefigure some of the results and some of the techniques that will appear slightly later in his 1954 paper "Quelques propriétés globales des variétés différentiables" and each of these notes contains a proof of the fact that $\Omega_3=0$. In fact, these two prior proofs are different from the demonstration that follows by specializing the results of his '54 paper, and which was alluded to in the above answers.

His '52 proof (in the 1953 proceedings) is close to Pontryagin's ideas relating framed cobordism groups to stable homotopy groups of spheres. Thom considers there the map $\psi_k:\pi_{n+k}(S^n) \to \Omega_k$ which consists in taking regular preimages of smooth approximations of maps $S^{n+k} \to S^n$. This map is additive and its image consists of the cobordism classes of those $k$-manifolds which can be embedded in $\mathbb{R}^{n+k}$ with trivial normal bundle. Since any closed orientable $3$-manifold is parallelizable, the map $\psi_3$ is surjective (for $n$ large enough) and it suffices to prove that $\psi_3$ vanishes on a generating set of the group $\pi_{n+3}(S^n)$. To conclude his argument, Thom mentions that this stable homotopy group is isomorphic to $\mathbb{Z}_{24}$ (which, I guess, he knew from Cartan or Serre) and, most importantly, that it is generated by the suspension of the quaternionic Hopf fibration with fiber $S^3= \partial D^4$.

In contrast with his '52 and '54 proofs, his '51 proof (in the 1952 proceedings) does not use any kind of Pontryagin-Thom constructions but, as Lickorish and Rourke will do some years later, he uses the fact that any closed oriented $3$-manifold can be presented by a Heegaard splitting. Let $H_g$ be the handlebody of genus $g$, let $\hbox{MCG}(\partial H_g)$ denote the mapping class group of its boundary and for any $f \in \hbox{MCG}(\partial H_g)$, set

$$ V_f := H_g \cup_f (-H_g). $$

Thom starts by observing that, if $V_f$ and $V_{f'}$ bound, then so does $V_{f \circ f'}$ (see Bruno's comment to Daniel's answer). Next, he recalls that Dehn found an explicit and finite system of generators $\{\tau_i\}$ for $\hbox{MCG}(\partial H_g)$ which consists of several Dehn twists: there are 2, 5 and 2g(g-1) such generators for $g=1$, $g=2$ and $g>2$ respectively. Thus, by the previous observation, it suffices to check that $V_{\tau_i}$ is a boundary for any $g\geq 1$ and for any $i$. For any $g>1$, it can be observed that each of the Dehn twists $\tau_i$ is performed along a curve which avoids a full (solid) handle of $H_g$: hence $V_{\tau_i}$ is the connected sum of $S^1 \times S^2$ with a $3$-manifold admitting a Heegaard splitting of genus $g-1$. Since such a connected sum can be realized in dimension $4$ by the attachement of a handle of index $1$, we are allowed to do an induction on the genus $g$. For $g=1$, the $3$-manifolds $V_{\tau_1}$ and $V_{\tau_2}$ are $S^3$ and $S^1 \times S^2$, which are obviously boundaries.

It seems that this early proof that $\Omega_3=0$ has been forgotten since then: this is probably due to the fact that the proceedings is not widely available and is written in French. Personally, I did not know it before I opened this proceedings in the Math' Library of Strasbourg a few months ago! Nonetheless, Haefliger mentions it in his survey paper of Thom's works (Publ. IHES 1988).

Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time:

1. "Quelques propriétés des variétés-bords", Colloque de Topologie de Strasbourg, 1951, no. V. La Bibliothèque Nationale et Universitaire de Strasbourg, 1952.

2. "Sur les variétés cobordantes", Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 7. La Bibliothèque Nationale et Universitaire de Strasbourg, 1953.

These two notes prefigure some of the results and some of the techniques that will appear slightly later in his 1954 paper "Quelques propriétés globales des variétés différentiables" and each of these notes contains a proof of the fact that $\Omega_3=0$. In fact, these two prior proofs are different from the demonstration that follows by specializing the results of his '54 paper, and which was alluded to in the above answers.

His '52 proof (in the 1953 proceedings) is close to Pontryagin's ideas relating framed cobordism groups to stable homotopy groups of spheres. Thom considers there the map $\psi_k:\pi_{n+k}(S^n) \to \Omega_k$ which consists in taking regular preimages of smooth approximations of maps $S^{n+k} \to S^n$. This map is additive and its image consists of the cobordism classes of those $k$-manifolds which can be embedded in $\mathbb{R}^{n+k}$ with trivial normal bundle. Since any closed orientable $3$-manifold is parallelizable, the map $\psi_3$ is surjective (for $n$ large enough) and it suffices to prove that $\psi_3$ vanishes on a generating set of the group $\pi_{n+3}(S^n)$. To conclude his argument, Thom mentions that this stable homotopy group is isomorphic to $\mathbb{Z}_{24}$ (which, I guess, he knew from Cartan or Serre) and, most importantly, that it is generated by the suspension of the quaternionic Hopf fibration with fiber $S^3= \partial D^4$.

In contrast with his '52 and '54 proofs, his '51 proof (in the 1952 proceedings) does not use any kind of Pontryagin-Thom constructions but, as Lickorish and Fenn will do some years later, he uses the fact that any closed oriented $3$-manifold can be presented by a Heegaard splitting. Let $H_g$ be the handlebody of genus $g$, let $\hbox{MCG}(\partial H_g)$ denote the mapping class group of its boundary and for any $f \in \hbox{MCG}(\partial H_g)$, set

$$ V_f := H_g \cup_f (-H_g). $$

Thom starts by observing that, if $V_f$ and $V_{f'}$ bound, then so does $V_{f \circ f'}$ (see Bruno's comment to Daniel's answer). Next, he recalls that Dehn found an explicit and finite system of generators $\{\tau_i\}$ for $\hbox{MCG}(\partial H_g)$ which consists of several Dehn twists: there are 2, 5 and 2g(g-1) such generators for $g=1$, $g=2$ and $g>2$ respectively. Thus, by the previous observation, it suffices to check that $V_{\tau_i}$ is a boundary for any $g\geq 1$ and for any $i$. For any $g>1$, it can be observed that each of the Dehn twists $\tau_i$ is performed along a curve which avoids a full (solid) handle of $H_g$: hence $V_{\tau_i}$ is the connected sum of $S^1 \times S^2$ with a $3$-manifold admitting a Heegaard splitting of genus $g-1$. Since such a connected sum can be realized in dimension $4$ by the attachement of a handle of index $1$, we are allowed to do an induction on the genus $g$. For $g=1$, the $3$-manifolds $V_{\tau_1}$ and $V_{\tau_2}$ are $S^3$ and $S^1 \times S^2$, which are obviously boundaries.

It seems that this early proof that $\Omega_3=0$ has been forgotten since then: this is probably due to the fact that the proceedings is not widely available and is written in french. Personally, I did not know it before I opened this proceedings in the Math' Library of Strasbourg a few months ago! Nonetheless, Haefliger mentions it in his survey paper of Thom's works (Publ. IHES 1988).

Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time:

1. "Quelques propriétés des variétés-bords", Colloque de Topologie de Strasbourg, 1951, no. V. La Bibliothèque Nationale et Universitaire de Strasbourg, 1952.

2. "Sur les variétés cobordantes", Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 7. La Bibliothèque Nationale et Universitaire de Strasbourg, 1953.

These two notes prefigure some of the results and some of the techniques that will appear slightly later in his 1954 paper "Quelques propriétés globales des variétés différentiables" and each of these notes contains a proof of the fact that $\Omega_3=0$. In fact, these two prior proofs are different from the demonstration that follows by specializing the results of his '54 paper, and which was alluded to in the above answers.

His '52 proof (in the 1953 proceedings) is close to Pontryagin's ideas relating framed cobordism groups to stable homotopy groups of spheres. Thom considers there the map $\psi_k:\pi_{n+k}(S^n) \to \Omega_k$ which consists in taking regular preimages of smooth approximations of maps $S^{n+k} \to S^n$. This map is additive and its image consists of the cobordism classes of those $k$-manifolds which can be embedded in $\mathbb{R}^{n+k}$ with trivial normal bundle. Since any closed orientable $3$-manifold is parallelizable, the map $\psi_3$ is surjective (for $n$ large enough) and it suffices to prove that $\psi_3$ vanishes on a generating set of the group $\pi_{n+3}(S^n)$. To conclude his argument, Thom mentions that this stable homotopy group is isomorphic to $\mathbb{Z}_{24}$ (which, I guess, he knew from Cartan or Serre) and, most importantly, that it is generated by the suspension of the quaternionic Hopf fibration with fiber $S^3= \partial D^4$.

In contrast with his '52 and '54 proofs, his '51 proof (in the 1952 proceedings) does not use any kind of Pontryagin-Thom constructions but, as Lickorish and Rourke will do some years later, he uses the fact that any closed oriented $3$-manifold can be presented by a Heegaard splitting. Let $H_g$ be the handlebody of genus $g$, let $\hbox{MCG}(\partial H_g)$ denote the mapping class group of its boundary and for any $f \in \hbox{MCG}(\partial H_g)$, set

$$ V_f := H_g \cup_f (-H_g). $$

Thom starts by observing that, if $V_f$ and $V_{f'}$ bound, then so does $V_{f \circ f'}$ (see Bruno's comment to Daniel's answer). Next, he recalls that Dehn found an explicit and finite system of generators $\{\tau_i\}$ for $\hbox{MCG}(\partial H_g)$ which consists of several Dehn twists: there are 2, 5 and 2g(g-1) such generators for $g=1$, $g=2$ and $g>2$ respectively. Thus, by the previous observation, it suffices to check that $V_{\tau_i}$ is a boundary for any $g\geq 1$ and for any $i$. For any $g>1$, it can be observed that each of the Dehn twists $\tau_i$ is performed along a curve which avoids a full (solid) handle of $H_g$: hence $V_{\tau_i}$ is the connected sum of $S^1 \times S^2$ with a $3$-manifold admitting a Heegaard splitting of genus $g-1$. Since such a connected sum can be realized in dimension $4$ by the attachement of a handle of index $1$, we are allowed to do an induction on the genus $g$. For $g=1$, the $3$-manifolds $V_{\tau_1}$ and $V_{\tau_2}$ are $S^3$ and $S^1 \times S^2$, which are obviously boundaries.

It seems that this early proof that $\Omega_3=0$ has been forgotten since then: this is probably due to the fact that the proceedings is not widely available and is written in french. Personally, I did not know it before I opened this proceedings in the Math' Library of Strasbourg a few months ago! Nonetheless, Haefliger mentions it in his survey paper of Thom's works (Publ. IHES 1988).