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K-theory gives a nice way to define vector bundles that don't actually exist. For example, given a singular variety $Y$ embedded into a smooth variety $X$ we can define the virtual normal bundle as $$ [N_{Y/X} ]:= [T_Y|_X] - [T_X] $$ This is useful for studying characteristic classes of singular spaces. What are other examples of virtual bundles and their applications?

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  • $\begingroup$ Fredholm index and orientability of moduli spaces. $\endgroup$ – Chris Gerig Jul 20 '17 at 21:56
  • $\begingroup$ @ChrisGerig Would you like to expand this comment into an answer with examples? $\endgroup$ – 54321user Jul 20 '17 at 21:57
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This shows up when trying to orient moduli spaces of objects, using the Fredholm index. In slightly more fancy language (for googling buzzwords), we try to equip a moduli space of objects (such as solutions to some PDE) with an orientation sheaf, using Quillen's construction of the determinant line bundle of a family of Fredholm operators. When you're in the right setup: you take your solution, look at the linearization of the PDE around that solution, and form its determinant line bundle, which has a $\mathbb{Z}/2\mathbb{Z}$ choice of orientation.

Here is a possibly simple example, which ultimately gives a homology orientation for the Seiberg-Witten invariants on a closed Riemannian 4-manifold $X$. Consider the (Fredholm) operator on differential forms $$\delta=d^\ast+d^+:\Omega^1(X)\to \Omega^0(X)\oplus\Omega^2_+(X)$$ Form the "determinant line bundle" $\text{det}(\delta)=\text{ker}(\delta)-\text{coker}(\delta)$ and look to orient this virtual bundle. If we can deform $\delta$ to a $\mathbb{C}$-linear operator $\delta_\ast$ that would be awesome, because $\text{det}(\delta_\ast)$ has a canonical orientation (it's built from $\mathbb{C}$-vector spaces). So a deformation $\delta_t$ from $\delta_0=\delta$ to $\delta_1=\delta_\ast$ (through Fredholm operators) induces an element $\text{det}(\delta_t)$ in the K-theory of the interval $[0,1]$. So this element is trivial and orientable, and as a result, an orientation associated with $\delta_\ast$ (which we know is $+1$) defines an orientation associated with $\delta$.

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  • $\begingroup$ I assume that the orientation sheaf that you get here is just the usual one from Verdier duality? As in a shift of the dualizing complex $t^! R$ for whatever ring of coefficients $R$ you want to use, and $t$ the terminal map to a one-point space. $\endgroup$ – ಠ_ಠ Jul 20 '17 at 23:42
  • $\begingroup$ I (un)fortunately don't know that language, I only know the basics from Bredon's Sheaf Theory. $\endgroup$ – Chris Gerig Jul 21 '17 at 0:05
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I don't really think of virtual vector bundles as negative vector bundles. One can prove that if $E$ and $F$ are vector bundles, that the $K$-theory classes $[E]=[F]$ if and only if $E\oplus \mathbb R^l\cong F\oplus \mathbb R^l$. Moreover, any $K$ theory class is of the form $[E]-[\mathbb R^l]$ for some $l$. Hence the $K$-theory classes are exactly the stable equivalence class of a vector bundle, possible shifted by dimension.

An explicit description of all vector bundles over a compact space is usually too hard, however the $K$-theory can sometimes be computed. This tells us exactly what the vector bundles are up to stable equivalence.

As Chris Gerig alludes to, the $K$-theory of a compact space is related to Fredholm operators by the Atiyah-Jänich theorem. Namely $K(X)\cong[X,\mathrm{F}(\mathbb H)]$, i.e. the $K$-theory groups can be seen as homotopy classes of maps into the Fredholm operators $F(\mathbb H)$. Elliptic PDE's define a map into the Fredholm operators, and this gives information on possible solution sets of the PDE.

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