In page 448 of these notes, a pairing between bordism and cobordism $$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$ is defined as follows. Assume $x\in U^m$ is represented by $\Sigma^{2l-m}X_+ \rightarrow MU(l) $ and $\alpha\in U_n$ is represented by $S^{2k+n}\rightarrow X_+\wedge MU(k)$. Compositing $\alpha$ and $x$, we get a map $$ S^{2k+2l+n-m}\rightarrow \Sigma^{2l-m}X_+\wedge MU(k)\rightarrow MU(l)\wedge MU(k)\rightarrow MU(l+k)$$ representing a class in $\Omega^U_{n-m}$, which is defined to be $\langle x,\alpha\rangle$.

In the following, the authors give a geometric intepretation of this pairing under certain assumptions on $x$ and $\alpha$. A screenshot for this argument can be found here. (I would upload the picture when I got 10 reputation points here, sorry for that!)

My questions are the following:

• I am trying to understand why the geometric pictures above correspond to the homotopical definition of the pairing. Intuitively, I could see that both the composition of maps and the intersection/pullback are both quite reasonable ways to define the pairing of $x$ and $\alpha$. But I haven't found out how to spell this out precisely.

• I also would like to know is there any known conditions on when can a cobordism class $x\in U^m(X)$ could be represented by an embedding $M\hookrightarrow X$ (when $0\leq m\leq \mathrm{dim}X $), or a fiber bundle $E\rightarrow X$ (when $m\leq 0$).

This is actually a question I posted on MSE but got no response so far. Thank you for your comments and answers!

In page 448 of these notes, a pairing between bordism and cobordism $$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$ is defined as follows. Assume $x\in U^m$ is represented by $\Sigma^{2l-m}X_+ \rightarrow MU(l) $ and $\alpha\in U_n$ is represented by $S^{2k+n}\rightarrow X_+\wedge MU(k)$. Compositing $\alpha$ and $x$, we get a map $$ S^{2k+2l+n-m}\rightarrow \Sigma^{2l-m}X_+\wedge MU(k)\rightarrow MU(l)\wedge MU(k)\rightarrow MU(l+k)$$ representing a class in $\Omega^U_{n-m}$, which is defined to be $\langle x,\alpha\rangle$.

In the following screenshot captured from the same page, the authors give a geometric intepretation of this pairing under certain assumptions on $x$ and $\alpha$:

My questions are the following:

• I am trying to understand why the geometric pictures above correspond to the homotopical definition of the pairing. Intuitively, I could see that both the composition of maps and the intersection/pullback are both quite reasonable ways to define the pairing of $x$ and $\alpha$. But I haven't found out how to spell this out precisely.

• I also would like to know is there any known conditions on when can a cobordism class $x\in U^m(X)$ could be represented by an embedding $M\hookrightarrow X$ (when $0\leq m\leq \mathrm{dim}X $), or a fiber bundle $E\rightarrow X$ (when $m\leq 0$).

This is actually a question I posted on MSE but got no response so far. Thank you for your comments and answers!

In page 448 of these notes, a pairing between bordism and cobordism $$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$ is defined as follows. Assume $x\in U^m$ is represented by $\Sigma^{2l-m}X_+ \rightarrow MU(l) $ and $\alpha\in U_n$ is represented by $S^{2k+n}\rightarrow X_+\wedge MU(k)$. Compositing $\alpha$ and $x$, we get a map $$ S^{2k+2l+n-m}\rightarrow \Sigma^{2l-m}X_+\wedge MU(k)\rightarrow MU(l)\wedge MU(k)\rightarrow MU(l+k)$$ representing a class in $\Omega^U_{n-m}$, which is defined to be $\langle x,\alpha\rangle$.

In the following, the authors give a geometric intepretation of this pairing under certain assumptions on $x$ and $\alpha$. A screenshot for this argument can be found here. (I would upload the picture when I got 10 reputation points here, sorry for that!)

My questions are the following:

• I am trying to understand why the geometric pictures above correspond to the homotopical definition of the pairing. Intuitively, I could see that both the composition of maps and the intersection/pullback are both quite reasonable ways to define the pairing of $x$ and $\alpha$. But I haven't found out how to spell this out precisely.

• I also would like to know is there any known conditions on when can a cobordism class $x\in U^m(X)$ could be represented by an embedding $M\hookrightarrow X$ (when $0\leq m\leq \mathrm{dim}X $), or a fiber bundle $E\rightarrow X$ (when $m\leq 0$).

This is actually a question I posted on MSE but got no response so far. Thank you for your comments and answers!

In page 448 of these notes, a pairing between bordism and cobordism $$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$ is defined as follows. Assume $x\in U^m$ is represented by $\Sigma^{2l-m}X_+ \rightarrow MU(l) $ and $\alpha\in U_n$ is represented by $S^{2k+n}\rightarrow X_+\wedge MU(k)$. Compositing $\alpha$ and $x$, we get a map $$ S^{2k+2l+n-m}\rightarrow \Sigma^{2l-m}X_+\wedge MU(k)\rightarrow MU(l)\wedge MU(k)\rightarrow MU(l+k)$$ representing a class in $\Omega^U_{n-m}$, which is defined to be $\langle x,\alpha\rangle$.

In the following, the authors give a geometric intepretation of this pairing under certain assumptions on $x$ and $\alpha$. A screenshot for this argument can be found here. (I would upload the picture when I got 10 reputation points here, sorry for that!)

My questions are the following:

• I am trying to understand why the geometric pictures above correspond to the homotopical definition of the pairing. Intuitively, I could see that both the composition of maps and the intersection/pullback are both quite reasonable ways to define the pairing of $x$ and $\alpha$. But I haven't found out how to spell this out precisely.

• I also would like to know is there any known conditions on when can a cobordism class $x\in U^m(X)$ could be represented by an embedding $M\hookrightarrow X$ (when $0\leq m\leq \mathrm{dim}X $), or a fiber bundle $E\rightarrow X$ (when $m\leq 0$).

This is actually a question I posted on MSE but got no response so far. Thank you for your comments and answers!