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The real KO coefficient table and eightfold periodicity

Statement

Assume AC. For n=0,…,7, the groups KO−n(pt) are respectively Z,Z/2,Z/2,0,Z,0,0,0. Multiplication by the volume-positive rank-eight spinor class β gives KOj(pt)≅KOj−8(pt) for every integer j. The degree-four integer generator is the rank-four quaternionic spinor image of the quaternionic line; the degree-eight generator is the actual symbol β.

Facts & Assumptions

Given: AC and the real bundle theory with its suspension grading.

[F1]

Independent stable clutching calculations give KO0(pt)=Z, KO−1=KO−2=Z/2, KO−3=0, and KSp0(pt)=Z, KSp−j(pt)=0 for 1≤j≤3 (Real and symplectic clutching groups in low degrees).

[F2]

The trivial rank-four Spin bundle induces the natural isomorphism KSp0(Y)→KO~0(Y+∧S4) with the specified balanced quaternionic spinor symbol (Real Spin Thom isomorphisms with real and quaternionic sources).

[F3]

Multiplication by the same rank-eight symbol β is an isomorphism on all finite pairs and integer degrees (Real Bott periodicity from the rank-eight Clifford–Thom map).

Proof technique: low clutching and the quaternionic rank-four boundary.

Proof

technique · direct construction
1.1F1

The first four groups are exactly [F1]. That proof uses components and the first two homotopy groups of stable O, independently of real Bott and of this table. Their integer and order-two generators are respectively the real line, the Möbius clutching and the one-turn stable orthogonal clutching.

1.2F2algebra

For pointed finite CW Y, the isomorphism [F2] commutes with basepoint evaluation and with its split section. Its map on the basepoint summand is also an isomorphism. Hence it restricts to an isomorphism on the two kernels, namely KSp~0(Y)≅KO~0(Y∧S4). Explicitly, any lift of a target-kernel element has zero basepoint restriction by injectivity on the basepoint group, which proves surjectivity on kernels; injectivity is inherited.

2.1F1F2step 1.2algebra

Taking Y=Sj∧S0 for j=0,1,2,3 in step 1.2 gives KO−4−j(pt)≅KSp−j(pt). The second list in [F1] therefore supplies Z,0,0,0 for degrees −4,−5,−6,−7. At j=0 the quaternionic line maps by the actual rank-four spinor symbol, fixing the stated generator. This uses the quaternionic source, not a false unshifted real rank-four orientation.

3.1F1F2F3step 1.1step 2.1algebra∎

Combining steps 1.1 and 2.1 proves all eight residue groups. The natural isomorphism [F3] transports them to every integer degree, with map multiplication by the fixed β. No complex Bott theorem, abstract periodicity of orthogonal groups or ABS comparison has entered this deduction. AC is carried from the exact suppliers, so no choice-free conclusion is asserted.

Depends on

Used by

Dependency tree · two levels

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Sources