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Homotopy orbits of representation spheres are Thom spaces

This short note explains why, given a representation \(\rho\) of a (possibly non-discrete) group \(G\), the following two spaces are homotopy equivalent:

  1. The homotopy orbits \((S^\rho)_{hG}\), where \(S^\rho\) is the one point compactification of the vector space on which \(\rho\) lives, and where the homotopy orbits are computed in based spaces.
  2. The Thom space of the vector bundle on \(BG\) corresponding to the representation \(\rho\).

This is basically just the Grothendieck construction (often a good way of computing both limits and colimits) plus a formula for the relation between colimits in based spaces and unbased spaces. (Throughout colimit means homotopy colimit.)

First consider \((S^\rho)_{hG}\) computed in unbased spaces. This is just the colimit of the functor \(BG \to \mathrm{Spaces}\), whose value at the unique object of \(BG\) is \(S^\rho\) (here \(BG\) denotes \(G\) thought of as a one-object (∞,1)-category —if \(G\) is discrete this is an ordinary category). To compute a colimit of a space valued functor you can alternatively take the Grothendieck construction and invert all coCartesian arrows. Since \(BG\) is a groupoid, these arrows come pre-inverted and we just need to take the total space.

I claim that the Grothendieck construction is just the fiberwise one-point compactification of the vector bundle on \(BG\) induced by \(\rho\). This is simply because the Grothendieck fibration is the pullback to \(BG\) of the universal bundle \(\mathrm{Spaces}_\ast \to \mathrm{Spaces}\), the fiber above the object corresponding to any space \(X\) thought of as an object of \(\mathrm{Spaces}\) is just \(X\) thought of as an ∞-groupoid.

OK, so the unbased \((S^\rho)_{hG}\) is the total space of the sphere bundle of \(\rho\). Recall that the Thom space is the quotient of this sphere bundle where you collapse the section of points at infinity to a single point. Why is the resulting of collapsing that section then the same colimit \((S^\rho)_{hG}\) but now computed in based spaces?

It’s because you compute colimits in based spaces as follows: if you have a diagram \(F : \mathcal{C} \to \mathrm{Spaces}_\ast\), its colimit is \(\mathrm{cofib} (\mathrm{colim}(\ast) \to \mathrm{colim}(UF))\) where \(U : \mathrm{Spaces}_\ast \to \mathrm{Spaces}\) is the forgetful functor. To prove this, just notice that you could define \(\mathrm{Spaces}_\ast\) as the subcategory of \(\mathrm{Arr}(\mathrm{Spaces})\) spanned by arrows where the domain is \(\ast\). The inclusion of \(\mathrm{Spaces}_\ast\) into the arrow category has a left adjoint which sends an arrow \(X \to Y\) to \(\mathrm{cofib} (X \to Y)\) (pointed by the image of \(X\) in the cofiber). So, to compute colimits in based spaces we can compute them in the arrow category (which is just a functor category so colimits are objectwise) and then take cofibers.

Omar Antolín Camarena