We show that the homological properties of a 5-manifold $M$ with fundamental group $G$ are encapsulated in a $G$-invariant stable form on the dual of the third syzygy of $mathbb{Z}$. In this notation one may express an even stronger version of Poincaré duality for $M$. However, we find an obstruction to this duality.
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机译：我们显示，具有基本群$ G $的5流形$ M $的同源性被封装在$ mathbb {Z} $的第三对偶的对偶中，以$ G $不变的稳定形式封装。用这种表示法可以表达$ M $的庞加莱对偶性的更强版本。但是，我们发现这种双重性受到阻碍。
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