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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
We state results on cohomological obstructions to and characterization of various G-structures on closed 8-manifolds.
(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:
The second and sixth Stiefel-Whitney classes (of the tangent bundle) vanish
The Euler class $\chi$ (of the tangent bundle) evaluated on $X$ (hence the Euler characteristic of $X$) is proportional to I8 evaluated on $X$:
The Euler characteristic is divisible by 4:
(Čadek-Vanžura 97, Corollary 5.5)
(Spin(4)-structure on 8-manifolds)
Let $X$ be a closed connected spin 8-manifold. Then $X$ has G-structure for $G =$ Spin(4)
if and only if the following conditions are satisfied:
the sixth Stiefel-Whitney class of the tangent bundle vanishes
the Euler class of the tangent bundle vanishes
the I8-term evaluated on $X$ is divisible as:
there exists an integer $k \in \mathbb{Z}$ such that
$p_2 = (2k - 1)^2 \left( \tfrac{1}{2} p_1 \right)^2$;
$\tfrac{1}{3} k (k+2) p_2[X] \;\in\; \mathbb{Z}$.
Moreover, in this case we have for $\widehat{T X}$ a given Spin(4)-structure as in (2) and setting
for $\chi_4$ the Euler class on $B Spin(4)$ (which is an integral class, by this Prop.)
the following relations:
$\tilde G_4$ (3) is an integer multiple of the first fractional Pontryagin class by the factor $k$ from above:
The (mod-2 reduction followed by) the Steenrod operation $Sq^2$ on $\widetilde G_4$ (3) vanishes:
the shifted square of $\tilde G_4$ (3) evaluated on $X$ is a multiple of 8:
The I8-term is related to the shifted square of $\widetilde G_4$ by
(Čadek-Vanžura 98a, Cor. 4.2 with Cor. 4.3)
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
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
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)
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)
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)
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