This is my first MO question...hopefully it's not a bad one...

Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands dirty I could look at the representing spaces or go through the Thom construction of $MU$.

I would like to give a talk aimed at a more general mathematical audience discussing how formal group laws and Quillen's Theorem get involved with the story of complex cobordism. To do this I'm going to want to introduce complex cobordism in the more classical way, e.g. following Poincare's original definition or the comment in Ravenel's *Complex Cobordism and Stable Homotopy* (page 10) that $MU_{*}(X)$ can be defined completely analogously to $H_{*}(X)$. So here's what the start of the talk would look like:

(1) Define $\Omega_n = \mathcal{M}^n/\sim$ where $\mathcal{M}^n$ is a particular collection of $n$-dimensional manifolds$^!$ and $\sim$ is the cobordism relation.

(2) Define $C_n(X) = \langle \sigma:M^n\rightarrow X \;|\; M^n\in \mathcal{M}^n \rangle /\sim$ where $\sim$ is the bordism relation. Then there must be some differential $C_n(X)\rightarrow C_{n-1}(X)$ which gives rise to a homology theory $MU_n(X)$ (or dually to $MU^n(X)$)

(3) Mention Thom's theorem that $\Omega_* \cong \pi_*(MU)$, and proceed from there.

My question is, how is that differential in (2) defined? Is it easy to show that $d^2 = 0$? More generally, do people think this is a reasonable way to introduce $MU$ to an audience containing no stable homotopy theorists?

$^!$: By manifold here I mean even dimensional real manifold with a map $J:TM\rightarrow TM$ such that $p\circ J \simeq p$ and $J^2 = -1$. So $J$ gives an $\mathbb{R}$-linear action of $i$ on $T_xM$ for all $x$. This seems to capture a larger class of manifolds than just complex manifolds, and I can't see any way to get a larger class than this.