# The cohomology of biquotients via a product on the two-sided bar construction (expository version)

@article{Carlson2021TheCO, title={The cohomology of biquotients via a product on the two-sided bar construction (expository version)}, author={Jeffrey D. Carlson}, journal={arXiv: Algebraic Topology}, year={2021} }

We compute the Borel equivariant cohomology ring of the left $K$-action on a homogeneous space $G/H$, where $G$ is a connected Lie group, $H$ and $K$ are closed, connected subgroups and $2$ and the torsion primes of the Lie groups are units of the coefficient ring. As a special case, this gives the singular cohomology rings of biquotients $H \backslash G / K$.
This depends on a version of the Eilenberg-Moore theorem developed in the appendix, where a novel multiplicative structure on the two…

## One Citation

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## References

SHOWING 1-10 OF 53 REFERENCES

### The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems

- Mathematics
- 1968

In Figure 1, Y-+B is an acyclic fibration with fibre F and E—>X is the fibration induced by f: X—>B. In Figure 2, ƒ: X—>B is a Serre fibration with fibre E. We assume (in both cases) that B is…

### The cohomology rings of homogeneous spaces

- MathematicsJournal of Topology
- 2021

Let G be a compact connected Lie group and K a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of G and K is invertible in a given principal ideal…

### Combinatorial operad actions on cochains

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2004

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to…

### On the topology of double coset manifolds

- Mathematics
- 1993

We consider a compact Lie group G and two closed subgroups H and K of G. The abstract product K x H operates on G by g. (k, h) = k lgh. If this operation happens to be free, the quotient K \ G / H is…

### The real and rational cohomology of differential fibre bundles

- Mathematics
- 1978

Consider a differential fibre bundle (E, ir, X, G/H, G). Under certain reasonable hypotheses, the cohomology of the total space E is computed in terms of the cohomology of the base space X and…

### Six model structures for DG-modules over DGAs: model category theory in homological action

- Mathematics
- 2014

In Part 1, we describe six projective-type model structures on the category of dierential graded modules over a dierential graded algebra A over a commutative ring R. When R is a eld, the six…

### The geometric realization of a Kan fibration is a Serre fibration

- Mathematics
- 1968

The object of this note is to prove the statement in the title which is asserted without proof in [1, Lemma 2.1]. We follow the terminology of [2, II, 3] except that a map of simplicial sets which is…

### On the homotopy-commutativity of groups and loop spaces

- Mathematics
- 1960

Introduction . It is well known that the loop space SI(B) in B is homotopy-commutative i f B is an H-space. Furthermore, it follows that the group G, which is also a CW-complex, is…

### On the cohomology of generalized homogeneous spaces

- Mathematics
- 2001

We observe that work of Gugenheim and May on the cohomology of classical homogeneous spaces G/H of Lie groups applies verbatim to the calculation of the cohomology of generalized homogeneous spaces…