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$$v_1$$-Periodic Homotopy Groups of $$SO(n)$$
Martin Bendersky, Hunter College, City University of New York, NY, and Donald M. Davis, Lehigh University, Bethlehem, PA
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Memoirs of the American Mathematical Society
2004; 90 pp; softcover
Volume: 172
ISBN-10: 0-8218-3589-0
ISBN-13: 978-0-8218-3589-0
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/172/815

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 $$\mathbf{Z}/2$$.

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.

Readership

Graduate students and research mathematicians interested in algebraic topology, manifolds, and cell complexes.

Table of Contents

• Introduction
• The BTSS of $${\rm BSpin}(n)$$ and the CTP
• Listing of results
• The 1-line of $${\rm Spin}(2n)$$
• Eta towers
• $$d_3$$ on eta towers
• Fine tuning
• Combinatorics
• Comparison with $$J$$-homology approach
• Proof of fibration theorem
• A small resolution for computing $${\rm ext}_{\mathcal A}$$
• Bibliography
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