nLab 8-manifold

Idea

A manifold of dimension 8.

Properties

Signature

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

(1)$$\sigma[X] \;=\; \tfrac{1}{45} \big( 7 p_2 - (p_1)^2 \big)[X] \,.$$

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 $X$ 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

   $$w_2 \;=\; 0 \,, \phantom{AAA} w_6 \;=\; 0$$

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

   $$\begin{aligned} 8 \chi[X] &= 192 \cdot I_8[X] \\ & = 4 \Big( p_2 - \tfrac{1}{2}\big(p_1\big)^2 \Big)[X] \end{aligned}$$

3. The Euler characteristic is divisible by 4:

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

(Čadek-Vanžura 97, Corollary 5.5)

Proposition

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

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

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

if and only if the following conditions are satisfied:

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

   $$w_6(T X) \;=\; 0$$

2. the Euler class of the tangent bundle vanishes

   $$\chi_8(T X) \;=\; 0$$

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

   $$\tfrac{1}{32} \Big( p_2 - \big( \tfrac{1}{2} \big( p_1 \big)^2 \big) \Big) \;\in\; \mathbb{Z}$$

4. there exists an integer $k \in \mathbb{Z}$ such that

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

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

Moreover, in this case we have for $\widehat{T X}$ a given Spin(4)-structure as in (2) and setting

(3)$$\widetilde G_4 \;\coloneqq\; \tfrac{1}{2} \chi_4(\widehat{T X}) + \tfrac{1}{4}p_1(T X)$$

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

the following relations:

1. $\tilde G_4$ (3) is an integer multiple of the first fractional Pontryagin class by the factor $k$ from above:

   $$\widetilde G_4 \;=\; k \cdot \tfrac{1}{2}p_1$$

2. The (mod-2 reduction followed by) the Steenrod operation $Sq^2$ on $\widetilde G_4$ (3) vanishes:

   $$Sq^2 \left( \widetilde G_4 \right) \;=\; 0$$

3. the shifted square of $\tilde G_4$ (3) evaluated on $X$ is a multiple of 8:

   $$\tfrac{1}{8} \left( \left( \widetilde G_4 \right)^2 - \widetilde G_4 \big( \tfrac{1}{2} p_1\big)[X] \right) \;\in\; \mathbb{Z}$$

4. The I8-term is related to the shifted square of $\widetilde G_4$ by

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

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

Examples

With exotic boundary 7-spheres

Consider $S^4$ the 4-sphere and let $D^4$ denote the 4-disk regarded as a manifold with boundary. Then a $D^4$-fiber bundle over $S^4$ is an 8-dimensional manifold with boundary.

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

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

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

If here $m+n = \pm 1$ 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

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

Here the left hand side must be an integer, while the right hand side is not an integer for all choices of pairs $(m,n)$. 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 $X$ with (exotic) 7-sphere boundaries correspond to near horizon limits of black M2 brane spacetimes $\mathbb{R}^{2,1} \times X$, 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)

Examples

• 8-sphere

• SU(3)

Related concepts

• M-theory on 8-manifolds

• Cayley 4-form

manifolds in low dimension:

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 5-manifolds

• 8-manifolds

References

General

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

$G$-Structure

• Martin Čadek, Jiří Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 (dml-cz:118764)

• Martin Čadek, Jiří Vanžura, On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

• Martin Čadek, Jiří Vanžura, On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes, Colloquium Mathematicum (1998), 76 (2), pp 213-228 (web)

• Martin Čadek, Jiří Vanžura, Almost quaternionic structures on eight-manifolds, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 (euclid:1200787905)

• Martin Čadek, Jiří Vanžura, Various structures in 8-dimensional vector bundles over 8-manifolds, Banach Center Publications (1998) Volume: 45, Issue: 1, page 183-197 (dml:208903)

• Martin Čadek, Michael Crabb, Jiří Vanžura, Obstruction theory on 8-manifolds, manuscripta math. 127 (2008), 167-186 (arXiv:0710.0734)

• Martin Čadek, Michael Crabb, Jiří Vanžura, Quaternionic structures, Topology and its Applications Volume 157, Issue 18, 1 December (2010), Pages 2850-2863 (doi:10.1016/j.topol.2010.09.005)

Exotic boundary 7-spheres

• Michael Joachim, D. J. Wraith, Exotic spheres and curvature (pdf)

• David Morrison, M. Ronen Plesser, section 3.2 of Non-Spherical Horizons, I, Adv. Theor. Math. Phys.3:1-81, 1999 (arXiv:hep-th/9810201)