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$v_1$-Periodic Homotopy Groups of $SO(n)$
 
Martin Bendersky Hunter College, City University of New York, New York, NY
Donald M. Davis Lehigh University, Bethlehem, PA
Front Cover for v_1-Periodic Homotopy Groups of SO(n)
Available Formats:
Electronic ISBN: 978-1-4704-0416-1
Product Code: MEMO/172/815.E
90 pp 
List Price: $63.00
MAA Member Price: $56.70
AMS Member Price: $37.80
Front Cover for v_1-Periodic Homotopy Groups of SO(n)
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  • Front Cover for v_1-Periodic Homotopy Groups of SO(n)
  • Back Cover for v_1-Periodic Homotopy Groups of SO(n)
$v_1$-Periodic Homotopy Groups of $SO(n)$
Martin Bendersky Hunter College, City University of New York, New York, NY
Donald M. Davis Lehigh University, Bethlehem, PA
Available Formats:
Electronic ISBN:  978-1-4704-0416-1
Product Code:  MEMO/172/815.E
90 pp 
List Price: $63.00
MAA Member Price: $56.70
AMS Member Price: $37.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1722004
    MSC: Primary 55; 57;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. The BTSS of BSpin($n$) and the CTP
    • 3. Listing of results
    • 4. The 1-line of Spin(2$n$)
    • 5. Eta towers
    • 6. $d_3$ on eta towers
    • 7. Fine tuning
    • 8. Combinatorics
    • 9. Comparison with $J$-homology approach
    • 10. Proof of fibration theorem
    • 11. A small resolution for computing $\operatorname {Ext}_{\mathcal {A}}$
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Volume: 1722004
MSC: Primary 55; 57;

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.

  • Chapters
  • 1. Introduction
  • 2. The BTSS of BSpin($n$) and the CTP
  • 3. Listing of results
  • 4. The 1-line of Spin(2$n$)
  • 5. Eta towers
  • 6. $d_3$ on eta towers
  • 7. Fine tuning
  • 8. Combinatorics
  • 9. Comparison with $J$-homology approach
  • 10. Proof of fibration theorem
  • 11. A small resolution for computing $\operatorname {Ext}_{\mathcal {A}}$
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