Return to Answer

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $\mathcal{N}^+_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^\infty MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $\mathcal{N}^+_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $\mathcal{N}^+_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^\infty MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $\mathcal{N}^+_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $\mathcal{N}^+_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^\infty MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $\mathcal{N}^+_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $\mathcal{N}^+_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^\infty MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $\mathcal{N}^+_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $N^{+}_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^{\infty} MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $N^{+}_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.

The Thom-Pontryagin construction is ingenious and unexpected, and turns out to be tremendously important. To construct an oriented cobordism group, consider the set of oriented compact $n$-manifolds, modulo the equivalence relation that two such manifolds are considered equivalent if they together bound an oriented compact $(n+1)$-manifold. This is seen to have the structure of an abelian group, where the inverse of a manifold is that manifold with the opposite orientation. The oriented cobordism group of points is isomorphic to $\mathbb{Z}$, where a pair of points oriented "+" and "-" cobound an oriented interval. The oriented cobordism group of circles is $\{0\}$, because any set of circles together cobound a surface, and the oriented cobordism group of compact surfaces is also $\{0\}$ because any oriented surface bounds a handlebody.

Cobordism groups had been in the air for a long time, with Poincaré having considered them as a first (failed) attempt to define homology. In general, there are an uncountable number of $n$--manifolds, so one would have expected oriented cobordism groups to be hopelessly untractable for large $n$. But, by a simple ingenious construction for which he won the Field's medal, René Thom showed that the oriented cobordism group $\mathcal{N}^+_{d-1}$ is isomorphic to $\pi_{d-1}\Omega^\infty MSO$, which are homotopy groups of the infinite loop space of the Thom spectrum. This is a very large space, but calculating its homotopy turns out to be quite simple, and it turns out to be the case that $\mathcal{N}^+_{d-1}$, modulo torsion is nothing more than a polynomial algebra $\mathbb{Q}[y_{4i}]_{i\geq 1}$ where $y_{4i}$ is a formal variable which can be represented by the complex projective space $\mathbb{CP}^{2i}$.

The Thom-Pontryagin construction itself is explained very nicely in an MO answer of Greg Kuperberg. A short lucid introduction to this story, on which this answer is loosely based, is given in the first few minutes of Ulrike Tillman's talk at the recent IMA conference in honour of John Milnor.