# A finite loop space not rationally equivalent to a compact Lie group

@article{Andersen2003AFL, title={A finite loop space not rationally equivalent to a compact Lie group}, author={Kasper K. S. Andersen and Tilman Bauer and Jesper Grodal and Erik Kjaer Pedersen}, journal={Inventiones mathematicae}, year={2003}, volume={157}, pages={1-10} }

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.

## 13 Citations

### RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

- MathematicsGlasgow Mathematical Journal
- 2021

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of…

### A torus theorem for homotopy nilpotent loop spaces

- Mathematics
- 2018

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces…

### A Torus Theorem for homotopy nilpotent groups

- Mathematics
- 2015

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces…

### Deconstructing Hopf spaces

- Mathematics
- 2004

We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We “deconstruct” the original space into an H-space Y with finite mod p cohomology and a finite…

### The Steenrod Problem of Classifying Polynomial Cohomology Rings

- Mathematics

In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question…

### The classification of p-compact groups and homotopical group theory

- Mathematics
- 2010

We survey some recent advances in the homotopy theory of classifying spaces, and homotopical group theory. We focus on the classification of p-compact groups in terms of root data over the p-adic…

### ON REALIZING RATIONAL AND POLYNOMIAL COHOMOLOGY RINGS

- Mathematics
- 2022

This is an expository paper written during the 2021 REU program at the University of Chicago. This paper focuses on the following problem: which graded-commutative R-algebras can be realized as the…

### The classification of p-compact groups for p odd

- Mathematics
- 2003

A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects…

### STABLY DUALIZABLE GROUPS

- Mathematics
- 2005

We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein [Kl01] and the p-complete study for p-compact groups by…

## References

SHOWING 1-10 OF 53 REFERENCES

### Centers and finite coverings of finite loop spaces.

- Mathematics
- 1994

A finite loop space X is a triple (X, BX, e), in which e : X -> ΩΒΧ is an equivalence from the space X into the loop space ΩΒΧ of the pointed space BX. The loop space X is called finite if X is…

### The finiteness obstruction for loop spaces

- Mathematics
- 1999

Abstract. For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness…

### Homotopy fixed-point methods for Lie groups and finite loop spaces

- Mathematics
- 1994

A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We…

### Finite loop spaces are manifolds

- Mathematics
- 2004

One of the motivating questions for surgery theory was whether every finite H:space is homotopy equivalent to a Lie group. This question was answered in the negative by Hilton and Roitberg 's…

### On π 3 of Finite Dimensional H-Spaces

- Mathematics
- 1963

THEOREM 1. If X is a pathwise connected and simply connected associative H-space of finite dimension and type, then either X is acyclic, i.e. Wq(X) 0 for all q, or there is at least one free summand…

### A new finite loop space at the prime two

- Mathematics, Physics
- 1993

We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We…

### Finite Unitary Reflection Groups

- MathematicsCanadian Journal of Mathematics
- 1954

Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of…