I hope to understand the following expression better:
$$w_3(V_{SO(3)}\otimes (TM-5)) = w_1^3(TM)+ w_3(V_{SO(3)}).$$
Here $V_{SO(3)}$ is the vector bundle of $SO(3)$. The $TM$ means the tangent bundle of a manifold $M$. The $TM-5$ means the construction of the tangent bundle and the virtual vector bundle.
The $V_1-V_2+V_3$ is an example of an operation on the vector bundles $V_1,V_2,V_3$ in the Grothendieck group. One way to define it, is as the isomorphism classes of ``virtual'' vector bundles, that is a pair of bundles $(V,V')$ modulo relation $(V\oplus W,V'\oplus W)\sim (V,V')$. The operation $+$ then descends from $\oplus$ so that $-(V,V')=(V',V)$. Then $V_i\equiv (V_i,0)$.
Question 1: Under what constraints of vector bundles, we can obtain the $w_3(V_{SO(3)}\otimes (TM-5))$ as the $w_1^3(TM)$ $+$ $w_3(V_{SO(3)})$?
Question 2: The $TM-5$ is a $0$-dimensional virtual vector bundle?
Question 3: How do we interpret the operations:
(1) the $\otimes$,
(2) the $+$,
(3) the $\oplus$,
(4) the $-$,
operations, precisely as the above?
Any useful references are welcome.