Abstract
We write (G,R) for a compact connected Lie group G of dimension d > 0 regarded as a framed manifold with the right invariant framing R and denote by [G,R] its bordism class determined in π d via the Thom-Pontrjagin construction. For this it is known [7] that 72[G,R] = 0 and more previously in [2] it is conjectured that [G,R] = 0 if rankG ≥ 10 or so. We have [SO(2n), R] = 0 (n ≥ 2) already in [3] and so we are interested in such a conjecture for the cases G = SO(n), SU(n) or Sp(n). In this note we consider a slight modification of Proposition 5.3 of [1] which describes the behavior of framings of G and using this we show the following partial results:
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