By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega_n^U(X)$ is represented by cocyles

$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$ where $M$ is a manifold with dimension $\dim X-n$, and the normal bundle of the first map is equipped with a complex structure.

I am wondering that if there is any way to turn each representative of $\Omega_n^U(X)$ into a

• inclusion $ M\hookrightarrow X$, when $n\geq0$,
• fibering $M\rightarrow X$, when $n\leq0$.

I know that we may turn a map into a inclusion or a fibering with the mapping cylinder/pathspace construction, however, this seems not working directly because of dimension reasons.

By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles

$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$ where $M$ is a manifold with dimension $\dim X-n$, and the normal bundle of the first map is equipped with a complex structure.

I am wondering that if there is any way to turn each representative of $\Omega^n_U(X)$ into a

• inclusion $ M\hookrightarrow X$, when $n\geq0$,
• fibering $M\rightarrow X$, when $n\leq0$.

I know that we may turn a map into a inclusion or a fibering with the mapping cylinder/pathspace construction, however, this seems not working directly because of dimension reasons.

By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega_n^U(X)$ is represented by cocyles

$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$ where $M$ is a manifold with dimension $\mathrm{dim}X-n$, and the normal bundle of the first map is equipped with a complex structure.

I am wondering that if there is any way to turn each representative of $\Omega_n^U(X)$ into a

• inclusion $ M\hookrightarrow X$, when $n\geq0$,
• fibering $M\rightarrow X$, when $n\leq0$.

I know that we may turn a map into a inclusion or a fibering with the mapping cylinder/pathspace construction, however, this seems not working directly because of dimension reasons.

Thank you very much for your comments and answers!

By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega_n^U(X)$ is represented by cocyles

$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$ where $M$ is a manifold with dimension $\dim X-n$, and the normal bundle of the first map is equipped with a complex structure.

I am wondering that if there is any way to turn each representative of $\Omega_n^U(X)$ into a

• inclusion $ M\hookrightarrow X$, when $n\geq0$,
• fibering $M\rightarrow X$, when $n\leq0$.

I know that we may turn a map into a inclusion or a fibering with the mapping cylinder/pathspace construction, however, this seems not working directly because of dimension reasons.