nLab
Borel-equivariant rational homotopy theory

Contents

Context

Rational homotopy theory

Representation theory

Contents

Idea

What may be called Borel-equivariant rational homotopy theory (this is not an established term, but it fits well) is the rational homotopy theory of G-spaces regarded in Borel equivariant homotopy theory, hence where weak equivalences are simply the underlying (forgetting the group action) weak homotopy equivalences. This is in contrast to the fixed locus-wise weak hom. equivalences, as in proper equivariant homotopy theory, which leads to proper equivariant rational homotopy theory.

Borel-equivariant rational homotopy theory is relevant even and particularly to the study of plain (non-equivariant) homotopy types, as it provides the means to conceptualize their rationalization beyond the constraint of these being nilpotent or even simply connected, hence outside the scope of plain rational homotopy theory:

Namely, given any connected space XX (but the same discussion applies to each connected component of a non-connected space), without any restriction on its fundamental group, it fits into a homotopy fiber sequence

X^ X Bπ 1(X) \array{ \widehat X &\xrightarrow{\;\;}& X \\ && \big\downarrow \\ && B \pi_1(X) }

with

Now, by the general discussion at \infty -action and at Borel model structure (here) – this means that XX regarded in the slice over BπB \pi is equivalent to the covering space X^\widehat X equipped with its canonical group action by π\pi.

This allows to regard the rationalization of non-nilpotent spaces XX as being the image of the universal covers X^\widehat X in π 1(X)\pi_1(X)-eqjuivariant homotopy theory. (Since the π 1(X)\pi_1(X)-action on a universal covering space is free, this is necessarily Borel-equivariant.)

This perspective is really due to Brown & Szczarba 95 (based on Brown & Szczarba 90, Brown & Szczarba 93) – to spot it there, look (here) for: 2nd par. on p. 883 (16 of 48) and around Thm. 4.4 on p. 896 (29 of 48).

References

The notion of fiberwise rationalization is due to

Rational homotopy theory (as well as real homotopy theory) for general connected spaces XX and regarded as Borel-equivariant rational homotopy theory of their covering spaces is laid out, and the corresponding fundamental theorem of dg-algebraic rational homotopy theory is proven, in:

based on

and further developed in:

A textbook account is in:

(While FHT 15 does discuss, mostly in its Section 7, some aspects of non-simply connected spaces via the rational homotopy of their covering spaces with attention to π 1\pi_1-actions, it does not seem to establish or mention aspects of a systematic generalization of the fundamental theorem of dg-algebraic rational homotopy theory to this situation – nor does it cite any of the above.)

Last revised on September 27, 2021 at 11:39:50. See the history of this page for a list of all contributions to it.