Corollary 9 in these notes by Ralph Cohen https://cmsa.fas.harvard.edu/wp-content/uploads/2022/03/immersions-revised2.pdf has grabbed my attention.

I do not undestand how to show that if we have a rank $k$ bundle which is stably isomorphic to the stable normal bundle then there is a virtual normal bundle of rank $k$.

This seems to boil down to proving the following thing:

Let $M^m$ a smooth manifold and suppose we are given $f:M^m\to BO(k)$. Let $n>k$ and let  $g: M^m\to BO(n)$ be such that $$TM\oplus g^*(EO(n)) \simeq \varepsilon^{n+m}$$ in other words, g is the classifying map for an virtual normal bundle of rank $n$ ($\varepsilon$ is the trivial bundle).

Suppose that $g$ is homotopic to $i\circ f$ where $i:BO(k)\to BO(n)$ is the obvious inclusion. Then we would like to show that  $$TM\oplus f^*(EO(k)) \simeq \varepsilon^{k+m}.$$

However the only thing I can conclude from the homotopy $g\simeq i\circ f$ is that $$TM\oplus f^*(EO(k))\oplus \varepsilon^{n-k} \simeq \varepsilon^{n+m}$$, which is not enough in general.

Corollary 9 in these notes by Ralph Cohen has grabbed my attention.

I do not undestand how to show that if we have a rank $k$ bundle which is stably isomorphic to the stable normal bundle then there is a virtual normal bundle of rank $k$.

This seems to boil down to proving the following thing:

Let $M^m$ a smooth manifold and suppose we are given $f:M^m\to BO(k)$. Let $n>k$ and let $g: M^m\to BO(n)$ be such that

$$TM\oplus g^*(EO(n)) \simeq \varepsilon^{n+m}.$$

In other words, $g$ is the classifying map for a virtual normal bundle of rank $n$ ($\varepsilon$ is the trivial bundle).

Suppose that $g$ is homotopic to $i\circ f$ where $i:BO(k)\to BO(n)$ is the obvious inclusion. Then we would like to show that 

$$TM\oplus f^*(EO(k)) \simeq \varepsilon^{k+m}.$$

However the only thing I can conclude from the homotopy $g\simeq i\circ f$ is that

$$TM\oplus f^*(EO(k))\oplus \varepsilon^{n-k} \simeq \varepsilon^{n+m},$$

which is not enough in general.