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nLab > Latest Changes: 8-manifold

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019

    am starting an entry here in order to record some facts. Not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019
    • (edited Mar 23rd 2019)

    added statement of this fact (here)

    Let X be a closed connected 8-manifold. Then X has G-structure for G= Spin(5) if and only if the following conditions are satisfied:

    1. The second and sixth Stiefel-Whitney classes (of the tangent bundle) vanish

      w2=0AAAw6=0

    2. The Euler class χ (of the tangent bundle) evaluated on X (hence the Euler characteristic of X) is proportional to I8 evaluated on X:

      8χ[X]=192⋅I8[X]=4(p2−12(p1)2)[X]

    3. The Euler characteristic is divisible by 4:

      χ[X]=0∈ℤ/4

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019
    • (edited Mar 23rd 2019)

    added statement of this fact (here):

    Let X be a closed connected spin 8-manifold. Then X has G-structure for G= Spin(4)

    BSpin(4)^TX↗↓X⟶TXBSpin(8)

    if and only if the following conditions are satisfied:

    1. the sixth Stiefel-Whitney class of the tangent bundle vanishes

      w6(TX)=0

    2. the Euler class of the tangent bundle vanishes

      χ8(TX)=0

    3. the I8-term evaluated on X is divisible as:

      132(p2−(12(p1)2))∈ℤ

    4. there exists an integer k∈ℤ such that

      1. p2=(2k−1)2(12p1)2;

      2. 13k(k+2)p2[X]∈ℤ.

    Moreover, in this case we have for ˆTX a given Spin(4)-structure as in (eq:Spin4Structure) and setting

    ˜G4≔12χ4(^TX)+14p1(TX)

    for χ4 the Euler class on BSpin(4) (which is an integral class, by this Prop.)

    the following relations:

    1. ˜G4 (eq:TildeG4) is an integer multiple of the first fractional Pontryagin class by the factor k from above:

      ˜G4=k⋅12p1

    2. The (mod-2 reduction followed by) the Steenrod operation Sq2 on ˜G4 (eq:TildeG4) vanishes:

      Sq2(˜G4)=0

    3. the shifted square of ˜G4 (eq:TildeG4) evaluated on X is a multiple of 8:

      18((˜G4)2−˜G4(12p1)[X])∈ℤ

    4. The I8-term is related to the shifted square of ˜G4 by

      $$ 4 \Big( \left( \widetilde G_4 \right)^2 - \widetilde G_4 \left( \tfrac{1}{2}p_1 \right) \Big) \;=\; \Big( p_2

      • \big( \tfrac{1}{2}p_1 \big)^2 \Big) $$

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2019
    • (edited Mar 23rd 2019)

    added a brief mentioning of 8-manifolds with exotic boundary 7-spheres (here) – so far just a glorified pointer to

    diff, v5, current

  5. Note that the signature of an 8-manifold need not necessarily be 1 or -1. Take for example the spheres S^{4k}, k a positive integer, which has signature 0.

    Cole Durham

    diff, v12, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2020
    • (edited Sep 9th 2020)

    Thanks for catching.

    (This statement was copy-and-pasted from discussion of Milnor’s construction of exotic 7-spheres, where the signature is ±1.)

    So I have fixed the actual formula now.

    diff, v13, current

  7. Added example section and added 8-sphere and SU(3).

    diff, v14, current

1 to 7 of 7

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