Rationally acyclic mean that the all rational homology groups are vanish except the zeroth homology group. Here closed manifold mean compact without boundary. In even dimension, we have nice examples of rationally acyclic closed connected manifolds. For example even dimensional real projective spaces. The rational homology groups of $\mathbb{RP}^{2n}$ are $H_{0}(\mathbb{RP}^{2n};\mathbb{Q})=\mathbb{Q}$ and $H_{i>0}(\mathbb{RP}^{2n};\mathbb{Q})=0.$ It is possible for closed odd dimensional manifold or not? In oriented case it is not possible because the top homology group of oriented closed manifold is non zero.
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$\begingroup$ As someone who knows nothing about manifolds, what is a rationally acyclic closed manifold? $\endgroup$– TyroneCommented Jun 24, 2018 at 8:51
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1 Answer
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No! Euler characteristics is an obstruction. Any closed odd dimensional manifold has $\chi=0$ [follows from Poincaré duality]. But if it is rationally acyclic, then $\chi=1$. Contradiction.
answered Jun 24, 2018 at 21:23
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5$\begingroup$ Nice! I spent a long time thinking about this to no avail - it looks like I was trying to be too fancy. $\endgroup$– user98602Commented Jun 24, 2018 at 21:58
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$\begingroup$ @MikeMiller I can understand. This happens...I was actually thinking about such 3 manifolds and all of a sudden this idea striked. $\endgroup$ Commented Jun 24, 2018 at 21:59