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    • 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 XX be a closed connected 8-manifold. Then XX has G-structure for G=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

      w 2=0AAAw 6=0 w_2 \;=\; 0 \phantom{AAA} w_6 \;=\; 0
    2. The Euler class χ\chi (of the tangent bundle) evaluated on XX (hence the Euler characteristic of XX) is proportional to I8 evaluated on XX:

      8χ[X] =192I 8[X] =4(p 212(p 1) 2)[X] \begin{aligned} 8 \chi[X] &= 192 \cdot I_8[X] \\ & = 4 \Big( p_2 - \tfrac{1}{2}\big(p_1\big)^2 \Big)[X] \end{aligned}
    3. The Euler characteristic is divisible by 4:

      χ[X]=0/4 \chi[X] \;=\; 0 \;\in\; \mathbb{Z}/4

    v1, current

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

    added statement of this fact (here):

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

    BSpin(4) TX^ X TX BSpin(8) \array{ && B Spin(4) \\ & {}^{\mathllap{ \widehat{T X} }} \nearrow & \big\downarrow \\ X & \underset{T X}{\longrightarrow} & B Spin(8) }

    if and only if the following conditions are satisfied:

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

      w 6(TX)=0 w_6(T X) \;=\; 0
    2. the Euler class of the tangent bundle vanishes

      χ 8(TX)=0 \chi_8(T X) \;=\; 0
    3. the I8-term evaluated on XX is divisible as:

      132(p 2(12(p 1) 2)) \tfrac{1}{32} \Big( p_2 - \big( \tfrac{1}{2} \big( p_1 \big)^2 \big) \Big) \;\in\; \mathbb{Z}
    4. there exists an integer kk \in \mathbb{Z} such that

      1. p 2=(2k1) 2(12p 1) 2p_2 = (2k - 1)^2 \left( \tfrac{1}{2} p_1 \right)^2;

      2. 13k(k+2)p 2[X]\tfrac{1}{3} k (k+2) p_2[X] \;\in\; \mathbb{Z}.

    Moreover, in this case we have for T^X\widehat T X a given Spin(4)-structure as in (eq:Spin4Structure) and setting

    G˜ 412χ 4(TX^)+14p 1(TX) \widetilde G_4 \;\coloneqq\; \tfrac{1}{2} \chi_4(\widehat{T X}) + \tfrac{1}{4}p_1(T X)

    for χ 4\chi_4 the Euler class on BSpin(4)B Spin(4) (which is an integral class, by this Prop.)

    the following relations:

    1. G˜ 4\tilde G_4 (eq:TildeG4) is an integer multiple of the first fractional Pontryagin class by the factor kk from above:

      G˜ 4=k12p 1 \widetilde G_4 \;=\; k \cdot \tfrac{1}{2}p_1
    2. The (mod-2 reduction followed by) the Steenrod operation Sq 2Sq^2 on G˜ 4\widetilde G_4 (eq:TildeG4) vanishes:

      Sq 2(G˜ 4)=0 Sq^2 \left( \widetilde G_4 \right) \;=\; 0
    3. the shifted square of G˜ 4\tilde G_4 (eq:TildeG4) evaluated on XX is a multiple of 8:

      18((G˜ 4) 2G˜ 4(12p 1)[X]) \tfrac{1}{8} \left( \left( \widetilde G_4 \right)^2 - \widetilde G_4 \big( \tfrac{1}{2} p_1\big)[X] \right) \;\in\; \mathbb{Z}
    4. The I8-term is related to the shifted square of G˜ 4\widetilde G_4 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

  1. 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\pm 1.)

    So I have fixed the actual formula now.

    diff, v13, current

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

    diff, v14, current