Electronic ISBN:  9781470404161 
Product Code:  MEMO/172/815.E 
List Price:  $63.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 172; 2004; 90 ppMSC: Primary 55; 57;
We compute the 2primary \(v_1\)periodic homotopy groups of the special orthogonal groups \(SO(n)\). The method is to calculate the BenderskyThompson 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.ReadershipGraduate 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 1line 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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We compute the 2primary \(v_1\)periodic homotopy groups of the special orthogonal groups \(SO(n)\). The method is to calculate the BenderskyThompson 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.
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 1line 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 $\operatorname {Ext}_{\mathcal {A}}$