Vb1s-periodic Homotopy Groups Of So(n (Memoirs of the American Mathematical Society)

Rok: 2004
ISBN: 9780821835890
OKCZID: 110057311

Citace (dle ČSN ISO 690):
BENDERSKY, Martin. v1-periodic homotopy groups of SO(n). Providence, R.I.: American Mathematical Society, 2004. vii, 90 s. Memoirs of the American Mathematical Society, no. 815.

Anotace

We compute the 2-primary $v_1$-periodic homotopy groups of the special orthogonal groups $SO(n)$. The method is to calculate the Bendersky-Thompson spectral sequence, a $K_*$-based unstable homotopy spectral sequence, of $\operatorname{Spin}(n)$. The $E_2$-term is an Ext group in a category of Adams modules. Most of the differentials in the spectral sequence are determined by naturality from those in the spheres. The resulting groups consist of two main parts. One is summands whose order depends on the minimal exponent of 2 in several sums of binomial coefficients times powers. The other is a sum of roughly $[\log_2(2n/3)]$ copies of ${\bold Z}/2$. As the spectral sequence converges to the $v_1$-periodic homotopy groups of the $K$-completion of a space, one important part of the proof is that the natural map from $\operatorname{Spin}(n)$ to its $K$-completion induces an isomorphism in $v_1$-periodic homotopy groups.

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