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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:
The second and sixth Stiefel-Whitney classes (of the tangent bundle) vanish
$w_2 \;=\; 0 \phantom{AAA} w_6 \;=\; 0$The Euler class $\chi$ (of the tangent bundle) evaluated on $X$ (hence the Euler characteristic of $X$) is proportional to I8 evaluated on $X$:
$\begin{aligned} 8 \chi[X] &= 192 \cdot I_8[X] \\ & = 4 \Big( p_2 - \tfrac{1}{2}\big(p_1\big)^2 \Big)[X] \end{aligned}$The Euler characteristic is divisible by 4:
$\chi[X] \;=\; 0 \;\in\; \mathbb{Z}/4$added statement of this fact (here):
Let $X$ be a closed connected spin 8-manifold. Then $X$ has G-structure for $G =$ Spin(4)
$\array{ && B Spin(4) \\ & {}^{\mathllap{ \widehat{T X} }} \nearrow & \big\downarrow \\ X & \underset{T X}{\longrightarrow} & B Spin(8) }$if and only if the following conditions are satisfied:
the sixth Stiefel-Whitney class of the tangent bundle vanishes
$w_6(T X) \;=\; 0$the Euler class of the tangent bundle vanishes
$\chi_8(T X) \;=\; 0$the I8-term evaluated on $X$ is divisible as:
$\tfrac{1}{32} \Big( p_2 - \big( \tfrac{1}{2} \big( p_1 \big)^2 \big) \Big) \;\in\; \mathbb{Z}$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 (eq:Spin4Structure) and setting
$\widetilde G_4 \;\coloneqq\; \tfrac{1}{2} \chi_4(\widehat{T X}) + \tfrac{1}{4}p_1(T X)$for $\chi_4$ the Euler class on $B Spin(4)$ (which is an integral class, by this Prop.)
the following relations:
$\tilde G_4$ (eq:TildeG4) is an integer multiple of the first fractional Pontryagin class by the factor $k$ from above:
$\widetilde G_4 \;=\; k \cdot \tfrac{1}{2}p_1$The (mod-2 reduction followed by) the Steenrod operation $Sq^2$ on $\widetilde G_4$ (eq:TildeG4) vanishes:
$Sq^2 \left( \widetilde G_4 \right) \;=\; 0$the shifted square of $\tilde G_4$ (eq:TildeG4) evaluated on $X$ is a multiple of 8:
$\tfrac{1}{8} \left( \left( \widetilde G_4 \right)^2 - \widetilde G_4 \big( \tfrac{1}{2} p_1\big)[X] \right) \;\in\; \mathbb{Z}$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
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