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

Abstract

Abstract. 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.

The classification of p-compact groups for p odd

“…(For another instance see [7].) Recall that a finite loop space is a loop space (X, BX, e), where X is a finite CW-complex.…”

Homotopy type and v1-periodic homotopy groups of p-compact groups

“…This is seen for X 3 from the entry in position (4, 3) The proof of Theorem 5.17 is entirely analogous, and is omitted.…”

“…See, e.g., Diagram 3.5. The 1 (3,2) of Theorem 4.18 and in the formula for ψ k (x 11 ) in [17,Proposition 3.5] tell that in both of our sequences, the determining element of π 38 (X 2 ) is a generator, i.e., it maps to α 2 ∈ π 38 (S 23 ), and so, by Theorem 3.7, there is a homotopy equivalence X 3 X 3 . This is probably the only place that any of our calculations with (X 31 ) 5 are required in proving Theorem 4.20.…”

“…In this case, the homotopy classes that must correspond are, in both cases, α 1 ∈ π 14 (S 7 ) (5) ≈ Z/5 and a generator of π 22 (B(7, 15)) ≈ Z/25, mapping to α 1 ∈ π 22 (S 15 ). The α 1 's in B 3 3 are well known by the action of P 1 , while in X 3 they may be deduced from entries in the matrix of Theorem 4.8 in positions (2, 1) and (3,2).…”

“…For example, we have 1,1,1,1,1) ; • f 12 = 136m (12) − 26 12 3 m (9,3) + 28 12 6 m (6,6) + (e)m e , where e ranges over (6,3,3), (3,3,3,3), (2, 2, 2, 2, 2, 2), (7, 1, 1, 1, 1, 1), (4, 4, 1, 1, 1, 1) ; (18) (15,3) (12,6) + 18 9 (1 + 27 2 )m (9,9) + (e)m e , where e ranges over (12,3,3), (9, 6, 3), (9, 3, 3, 3), (6,6,6), (6,6,3,3), (6, 3, 3, 3, 3), (3, 3, 3, 3, 3, 3), 1, 1, 1, 1, 1), (10, 4, 1, 1, 1, 1), (7, 4, 4, 1, 1, 1), (4, 4, 4, 4, 1, 1), (7, 7, 1, 1, 1, 1), (8, 2, 2, 2, 2, 2), (5, 5, 2, 2, 2, 2) .…”