Reference Request: Thom Spectrum of a virtual vector bundle Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse.
I unsuccessfully browsed through literature to find a reference for an explicit construction for $\mathbb{T}h(-\xi)$ as elementary as possible, especially without deep knowledge in stable homotopy theory or much overload.
 A: In the meanwhile, I finally figured out a possible construction, which I believe to produce the pursued result.
Take a virtual vector bundle in the sense of a homotopy class of a map $$f\colon X\rightarrow \mathbb{Z}\times BO$$ with constant rank $n\in\mathbb{Z}$. (The rank of a virtual bundle in that setting is the map obtained by composing $f$ with the projection on the first factor.) Define a convergent filtration $$X_{-n}\subseteq X_{-n+1}\subseteq...\subseteq X$$via $$X_i\colon=f^{-1}(\{n\}\times BO_{n+i})$$ and denote by $E_i\colon=f^*\gamma_{i+n}$ the vector bundle over $X_i$ obtained by pulling back the universal (i+n)-dimensional vector bundle. Then it holds $E_{i+1}|_{X_i}\cong E_i\oplus \mathbb{R}$ and now one can go on as usal to obtain a spectrum out of the sequence $E_i$, i.e. the (i)-th space of the spectrum $Th(f)$ is $E_i$ and the structure maps are obtained by the isomorphisms mentioned above.
One can then show that the spectrum depends only on the homotopy class of $f$ up to equivalence.
If $f$ is an honest (n)-dimensional vector bundle this reproduces the usual construction of the Thom spectrum of a vector bundle, which is equivalent to the suspenion spectrum of the Thom space associated to the bundle. If $f\colon X\rightarrow BO_n$ is a classifying map for the bundle the map one composes f with the inclusion $BO_n\rightarrow BO$ and the canonical nontrivial involution $\iota\colon BO\rightarrow BO$ of $BO$ to get a map $$X\rightarrow \{-n\}\times BO\rightarrow \mathbb{Z}\times BO,$$ which represents the inverse bundle of $f$. The construction I sketched above with $\tilde{f}$ results in the spectrum $Th(-f)$ is originally desired.
