As the title suggests, I'm trying to find motivation on the definition of the Thom spectrum of $-\xi$, or more generally on the definition of the Thom spectrum of a virtual bundle.

In this paper by S. Bauer (middle of page 7) he defines the Thom spectrum for a virtual bundle of the form $\xi - \Bbb R^m\times X$ as the desuspension: $$ Th(\xi-\Bbb R^m\times X) = \sum^{-m}\Bbb S\wedge X^{\xi}$$

with this definition, let $\eta$ be a bundle over $X$ such that $\xi\oplus \eta\cong X\times \Bbb R^N$, then we have we have that $$Th(-\xi) = \sum^{-N}\Bbb S\wedge X^{\eta}$$

I'm wondering what is the reason of this definition and if it was somehow related with the concept of duality morphism (Rudyak, page 47, def 2.3)

My intuition is that with this definition we should have that the two spectra are dual in the sense above. This would explain for example why Bauer in the paper mentioned above, or Crabb & Knapp in this paper (page 90, Lemma 1.1) claim that there is a pairing given by cap product: $$\widetilde{h}^0(X^{-\xi})\times \widetilde{h}^r(X^{\xi})\to \widetilde{h}^r(X)$$ since we could interpret the first one as an homology group and then cap product is well defined.

So I tried proving that we have this aforementioned duality. In fact we have something resembling a duality between $Th(-\xi)$ and $Th(\xi)$: $$Th(-\xi)\wedge Th(\xi)=\sum^{-N}\Bbb S\wedge X^{\eta}\wedge \Bbb S\wedge X^{\xi}$$ $$ = \sum^{-N}\Bbb S\wedge X^{\eta}\wedge X^{\xi}$$ $$= \sum^{-N}\Bbb S\wedge (X\times X)^{\eta \times \xi}$$

but my problem is that $(X\times X)^{\eta \times \xi}$ is not quite $X_+\wedge \sum^N\Bbb S$, instead we have $$X_+\wedge \sum^N\Bbb S = \Delta^*(X\times X)^{\eta \times \xi}\wedge \Bbb S$$ where $\Delta \colon X \to X\times X$ is the diagonal map, and this prevents me for proving that we have an actual duality.

I'm slightly confused here so I apologise in advance if something is not super precise.