2026-03-03 00:00:00:03014313110http://paper.people.com.cn/rmrb/pc/content/202603/03/content_30143131.htmlhttp://paper.people.com.cn/rmrb/pad/content/202603/03/content_30143131.html11921 本版责编：任飞帆
Nature, Published online: 27 February 2026; doi:10.1038/d41586-026-00505-z
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The fundamental group of \(X\) at \(x_0\) is \[\pi_1(X, x_0) \;:=\; \bigl\{[\gamma] \mid \gamma \text{ is a loop based at } x_0\bigr\}\] equipped with the group operation of concatenation: \[[\gamma] \cdot [\delta] := [\gamma * \delta], \qquad (\gamma * \delta)(s) := \begin{cases} \gamma(2s) & s \in [0,\tfrac{1}{2}] \\ \delta(2s-1) & s \in [\tfrac{1}{2},1] \end{cases}\] The identity element is the class of the constant loop \([c_{x_0}]\), and the inverse of \([\gamma]\) is \([\bar\gamma]\) where \(\bar\gamma(s) := \gamma(1-s)\).
Or, as OpenAI put it on X, “We heard your feedback loud and clear, and 5.3 Instant reduces the cringe.”
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