nLab Borel-equivariant rational homotopy theory

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 $X$ (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

$$\array{ \widehat X &\longrightarrow& X \\ && \big\downarrow \\ && B \pi_1(X) }$$

with

• its universal cover $\widehat X$, which is simply connected;

• the classifying space $B \pi_1(X)$ of its fundamental group $\pi_1(X)$.

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

This allows to regard the rationalization of non-nilpotent spaces $X$ as being the image of the universal covers $\widehat X$ in $\pi_1(X)$-eqjuivariant homotopy theory. (Since the $\pi_1(X)$-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).

Related concepts

• twisted equivariant Chern character

• equivariant de Rham complex

References

The notion of fiberwise rationalization is due to

• Aldridge Bousfield, Daniel Kan, Section I.8 in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics Vol. 304, Springer 1972 (doi:10.1007/978-3-540-38117-4)

Rational homotopy theory (as well as real homotopy theory) for general connected spaces $X$ 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:

• Edgar Brown, Robert H. Szczarba, Real and Rational Homotopy Theory, Chapter 17 in: Handbook of Algebraic Topology, North Holland (1995) 869-915 (doi:10.1016/B978-044481779-2/50018-3, ZB)

  • see p. 883 & 896 (16 & 29 of 48): pdf

based on

• Stephen Halperin, Daniel Tanré, Homotopie filtrée et fibres $C^\infty$, Illinois J. Math. 34(2): 284-324 (1990) (doi:10.1215/ijm/1255988268)

• Edgar Brown, Robert H. Szczarba, Continuous cohomology and Real homotopy type II Asterisque 191, Societe Mathematique De France (1990) (numdam:AST_1990__191__45_0)

• Edgar Brown, Robert H. Szczarba, Real and rational homotopy theory for spaces with arbitrary fundamental group, Duke Mathematical Journal 71. (1993): 229-316 (doi:10.1215/S0012-7094-93-07111-6)

with related discussion in

• Antonio Gómez-Tato, Stephen Halperin, Daniel Tanré, Rational homotopy theory for non-simply connected spaces, Trans. Amer. Math. Soc. 352 (2000), 1493-1525 (doi:10.1090/S0002-9947-99-02463-0, jstor:118074)

which is reviewed in:

• Sergei O. Ivanov, Section 12 of: An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups [arXiv:2111.10694]

A textbook account is in:

• Yves Félix, Steve Halperin, Jean-Claude Thomas, Rational Homotopy Theory II, World Scientific 2015 (doi:10.1142/9473)

(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 $\pi_1$-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.)

Further discussion of fiberwise rational homotopy theory:

• Manuel Rivera, Felix Wierstra, Mahmoud Zeinalian, Thm. 16 of: Rational homotopy equivalences and singular chains, Algebr. Geom. Topol. 21 (2021) 1535-1552 [arXiv:1906.03655. doi:10.2140/agt.2021.21.1535]