Todd genus as homomorphism from complex cobordism It's well-known that the Todd genus/arithmetic genus $\chi(\mathcal{O}_X)$ (or probably preferably $\int_X \text{td}(T_X)$ so as to define it in purely terms of the complex structure) is a genus in the sense of a homomorphism from complex cobordism. 
But I'm finding this difficult to reconcile this with my belief that in the cobordism ring, $[\mathbb{CP}^1]+[\mathbb{CP}^1]=[\mathbb{CP}^1\#\mathbb{CP}^1]=[\mathbb{CP}^1]$ - that is, two disjoint Riemann spheres are cobordant to the Riemann sphere's connected sum (gluing complex structures and all) with itself, which is isomorphic as a complex manifold to the Riemann sphere. But obviously the arithmetic genus of the complex projective line is not $0$, so this can't hold. Which part of this reasoning am I getting wrong?
 A: First of all, let me falsify $[\mathbb{CP}^1]+[\mathbb{CP}^1]=[\mathbb{CP}^1\#\mathbb{CP}^1]=[\mathbb{CP}^1]$ by a sledgehammer argument: The complex bordism ring $\Omega_*^U\cong \mathbb{Z}[x_{2i}|i\in\mathbb{N}]$ is an integral polynomial ring with one generator in each even dimension, so in particular has no torsion and $[X]+[X]=[X]$ can only hold for the trivial class. (The generator in degree 2 can even be chosen to be $\mathbb{CP}^1$.)
Where did you go right?


*

*$X\amalg Y$ is indeed complex bordant to $X\#Y$, so $[X]+[Y]=[X\amalg Y]=[X\#Y]$ holds.

*$S^2\cong\mathbb{CP}^1$ has a unique almost complex structure.


So what did go wrong?
Complex bordism classes are not necessarily represented by manifolds with an almost complex structure on the tangent bundle, but by manifolds with a tangential stably almost complex structure on their tangent bundle, which is induced by an isomorphism of the tangent bundle to a complex vector bundle after adding a trivial bundle, i.e. $TM\oplus\mathbb{R}^k\cong \xi$, where $\xi$ is a complex vector bundle and $k\in\mathbb{N}_0$, see this page on the manifold atlas. The construction of the complex bordism class $X\#Y$ produces such a tangential stably almost complex structure, which does not have to come from a honest almost complex structure on the tangent bundle, even if the structure on $X$ and $Y$ were induced by almost complex structures.
