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8-manifold

Contents

Contents

Idea

A manifold of dimension 8.

Properties

Signature

Let XX be a compact oriented smooth 8-manifold. Then its signature is related to the second Pontryagin class p 2p_2 and the cup product of the first Pontryagin class p 1p_1 with itself, both evaluated on the fundamental class of XX, by

(1)±1=σ[X]=145(7p 2(p 1) 2)[X]. \pm 1 \;=\; \sigma[X] \;=\; \tfrac{1}{45} \big( 7 p_2 - (p_1)^2 \big)[X] \,.

But in addition, due to the dimenion 8, the signature is ±1\pm 1, depending on the choice of orientation.

(e.g. Joachim-Wraith, p. 2)

G-Structures on 8-Manifolds

We state results on cohomological obstructions to and characterization of various G-structures on closed 8-manifolds.

Proposition

(Spin(5)-structure on 8-manifolds)

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=0,AAAw 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:

    14χ[X]. \tfrac{1}{4}\chi[X] \;\in\; \mathbb{Z} \,.

(Čadek-Vanžura 97, Corollary 5.5)


Proposition

(Spin(4)-structure on 8-manifolds)

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

(2) 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 TX^\widehat{T X} a given Spin(4)-structure as in (2) and setting

(3)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 (3) 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 (3) vanishes:

    Sq 2(G˜ 4)=0 Sq^2 \left( \widetilde G_4 \right) \;=\; 0
  3. the shifted square of G˜ 4\tilde G_4 (3) 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((G˜ 4) 2G˜ 4(12p 1))=(p 2(12p 1) 2) 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)

(Čadek-Vanžura 98a, Cor. 4.2 with Cor. 4.3)


Examples

With exotic boundary 7-spheres

Consider S 4S^4 the 4-sphere and D 4D^4 be the 4-disk, the latter a manifold with boundary. Then a D 4D^4-fiber bundle over S 4S^4 is an 8-dimensional manifold with boundary.

By the clutching construction, such bundles are classified by homotopy classes of maps

f (m,n):S 3SO(4) f_{(m,n)} \;\colon\; S^3 \longrightarrow SO(4)

from a 3-sphere (the equator of S 4S^4) to SO(4). By this Prop. such maps are classified by pairs of integers (m,n)×(m,n) \in \mathbb{Z} \times \mathbb{Z}.

If here m+n=±1m+n = \pm 1 then the boundary of the corresponding 8-manifold is homotopy equivalent to a 7-sphere, and in fact homeomorphic to the 7-sphere.

Assuming this 8-manifold is a smooth manifold, then plugging in the numbers into the signature formula (1) yields the relation

p 2[X]17(4(2m1) 2+45) p_2[X] \;\coloneqq\; \tfrac{1}{7} \big( 4(2m -1)^2 + 45 \big)

Here the left hand side must be an integer, which the right hand side is not an integer for all choices of pairs (m,n)(m,n). This means that for these choices the boundary 7-sphere is not diffeomorphic to the standard smooth 7-sphere – it is instead an exotic 7-sphere.

(see Joachim-Wraith, p. 2-3)

From the point of view of M-theory on 8-manifolds, these 8-manifolds XX with (exotic) 7-sphere boundaries correspond to near horizon limits of black M2 brane spacetimes 2,1×X\mathbb{R}^{2,1} \times X, where the M2-branes themselves would be sitting at the center of the 7-spheres (if that were included in the spacetime, see also Dirac charge quantization).

(Morrison-Plesser 99, section 3.2)


manifolds in low dimension:

References

General

  • Anand Dessai, Topology of positively curved 8-dimensional manifolds with symmetry (arXiv:0811.1034)

GG-Structure

Exotic boundary 7-spheres

Last revised on April 27, 2019 at 08:17:21. See the history of this page for a list of all contributions to it.