Equivalence of tangential and normal stably almost complex structure

Question

Let $M$ be a smooth manifold. $M$ is said to be tangential stably almost complex if $TM \oplus \underline{\mathbb{R}}^k$ can be given a structure of a complex vector bundle, for some $k$. $M$ is said to be normal stably almost complex if $\nu \oplus \underline{\mathbb{R}}^k$ can be given a structure of a complex vector bundle, where $\nu$ is the normal bundle of some embedding of $M$ into $\mathbb{R}^N$, for some $k$.

It is stated in a few places that tangential stably almost complex structures and normal stably almost complex structures are equivalent, see for example: http://www.map.mpim-bonn.mpg.de/Complex_bordism#Stably_complex_structures. However I cannot figure out why. In the above link, it is stated that this follows from $TM \oplus \nu = \underline{\mathbb{R}}^N$. So suppose I have a tangential stably almost complex structure, then $(TM \oplus \underline{\mathbb{R}}^k) \oplus \nu = \underline{\mathbb{R}}^{N+k}$, the first summand on the left has an almost complex structure and I can see that the right hand side can be given an almost complex structure as well. But then how does one proceed next?

It seems to me that one needs some method to extend an almost complex structure on $\mathbb{R}^m$ (the first summand) to $\mathbb{R}^{N+k}$ on each fibre, so that we can restrict it back to $\nu$. But since the space of almost complex structures $O(2n)/U(n)$ has nontrivial topology, whether this can be done seems to depend on the topology of $M$. So I guess I might be on the wrong path here. Any help is appreciated!

Answer 1

This would be my attempt to put a stable almost complex structure on $\nu$, given one on $TM$. Fix an almost complex structure on $TM\oplus \underline{\mathbb{R}}^k$.

The vector bundle $\nu\oplus \nu$ has a complex structure: namely define $J(v,w)=(w,-v)$, it then follows that $J^2(v,w)=(-v,-w)$, i.e. $\nu\oplus \nu$ is almost complex.

Look at the vector bundle $$ (\nu\oplus \nu)\oplus (TM\oplus \underline{\mathbb{R}}^k) $$

This has an almost complex structure (just define it on the summands separately). As a real bundle this is nothing but $$ (\nu\oplus \nu)\oplus (TM\oplus \underline{\mathbb{R}}^k)\cong\nu\oplus (\nu\oplus TM)\oplus \underline{\mathbb{R}}^k\cong \nu\oplus \underline{\mathbb{R}}^{N+k}. $$

Thus we have defined an stable almost complex structure on $\nu$.