On framed cobordism classes of classical Lie groups. (English) Zbl 1042.55004

Minami, Haruo

J. Math. Soc. Japan 55, No. 4, 1033-1052 (2003).

A compact connected Lie group \(G\) with left invariant framing \(L\) defines via the Thom-Pontrjagin construction an element \([G,L]\) in the stable homotopy groups of spheres. In this paper the problem is considered of determining whether or not this is zero when \(G\) is a classical Lie group. Such a problem has been studied by many authors. This work is especially motivated by the result by E. Ossa [Topology 21, 315–323 (1982; Zbl 0491.55008)] stating that \(24[G,L]=0\). The result proved here is: \([SO(2n+1), L]_{(3)}=0\) and \([Sp(n),L]_{(3)}=0\) for any \(n\geq 3\), \(n\neq 5\), 7, 11 where the subscript (3) indicates the 3-component of \([G,L]\). One previously knows that \([SO(2n),L]_{(3)}=0\) \((n\geq 1)\) and \([SU(n),L]_{(3)}=0\) \((n\geq 3)\) and also \([SO(3),L]\) and \([SO(5), L]\) generate the 3-components of the 3- and 10-dimensional stable homotopy groups of spheres, respectively. So it turns out that only the six ones above remain unknown in the classical case. The proof uses a theorem of J. C. Becker and R. Schultz [Fixed point indices and left invariant framings, Springer Lecture Notes 657, 1–31 (1977; Zbl 0391.55014)] and an idea of E. Ossa for the proof of the result mentioned above. Also, the author’s proof requires a formula of B. Steer [Topology 15, 383–393 (1976; Zbl 0336.57011)] concerning framings twisted by representations.

Reviewer: Haruo Minami (Nara)

MSC:

55N22	Bordism and cobordism theories and formal group laws in algebraic topology
19L20	\(J\)-homomorphism, Adams operations
57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)

Keywords:

framed manifold; classical Lie group; left invariant framing; \(J\)-morphism

Citations:

Zbl 0391.55014; Zbl 0336.57011; Zbl 0491.55008