The super-classical example would be the use of the (?Serre's?) theorem that $H^n(X;G) = [X,K(G,n)]$ to deduce that co-dimension two knots have Seifert surfaces. This is written up in Cameron Gordon's LNM 685 "Some aspects of classical knot theory".

The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^1 \partial C$ is either $\mathbb Z^2$ or $\mathbb Z$ according to whether or not $n=3$ or $n>3$. In either case the restriction map is an injection. Serre's theorem gives you a map $C \to S^1$ which you make transverse to a point (and a standard projection map on the boundary), this makes the preimage of this point a Seifert surface for the knot. By Seifert-surface I mean an orientable co-dimension one submanifold of $S^n$ whose boundary is the knot.

Serre and Thom used these ideas repeatedly in their early attacks on the Steenrod realization problem. Well, the version where you're trying to realize the homology classes by embedded submanifolds. This is a relative version of their arguments.

The super-classical example would be the use of the (?Serre's?) theorem that $H^n(X;G) = [X,K(G,n)]$ to deduce that co-dimension two knots have Seifert surfaces. This is written up in Kervaire and Weber's article in LNM 685 "A survey of multi-dimensional knots".

The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^1 \partial C$ is either $\mathbb Z^2$ or $\mathbb Z$ according to whether or not $n=3$ or $n>3$. In either case the restriction map is an injection. Serre's theorem gives you a map $C \to S^1$ which you make transverse to a point (and a standard projection map on the boundary), this makes the preimage of this point a Seifert surface for the knot. By Seifert-surface I mean an orientable co-dimension one submanifold of $S^n$ whose boundary is the knot.

Serre and Thom used these ideas repeatedly in their early attacks on the Steenrod realization problem. Well, the version where you're trying to realize the homology classes by embedded submanifolds. This is a relative version of their arguments.

The super-classical example would be the use of the (?Serre's?) theorem that $H^n(X;G) = [X,K(G,n)]$ to deduce that co-dimension two knots have Seifert surfaces. This is written up in Cameron Gordon's LNM 685 "Some aspects of classical knot theory".

The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^1 \partial C$ is either $\mathbb Z^2$ or $\mathbb Z$ according to whether or not $n=3$ or $n>3$. In either case the restriction map is an injection. Serre's theorem gives you a map $C \to S^1$ which you make transverse to a point (and a standard projection map on the boundary), this makes the preimage of this point a Seifert surface for the knot. By Seifert-surface I mean an orientable co-dimension one submanifold of $S^n$ whose boundary is the knot.

Serre and Thom used these ideas repeatedly in their early attacks on the Steenrod realization problem.

The super-classical example would be the use of the (?Serre's?) theorem that $H^n(X;G) = [X,K(G,n)]$ to deduce that co-dimension two knots have Seifert surfaces. This is written up in Cameron Gordon's LNM 685 "Some aspects of classical knot theory".

The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^1 \partial C$ is either $\mathbb Z^2$ or $\mathbb Z$ according to whether or not $n=3$ or $n>3$. In either case the restriction map is an injection. Serre's theorem gives you a map $C \to S^1$ which you make transverse to a point (and a standard projection map on the boundary), this makes the preimage of this point a Seifert surface for the knot. By Seifert-surface I mean an orientable co-dimension one submanifold of $S^n$ whose boundary is the knot.

Serre and Thom used these ideas repeatedly in their early attacks on the Steenrod realization problem. Well, the version where you're trying to realize the homology classes by embedded submanifolds. This is a relative version of their arguments.