Surfaces So Different Even a Fourth Dimension Can’t Make Them the Same

In geometry and the closely related field of topology, adding a spatial dimension can often have wondrous effects: Previously distinct objects become indistinguishable. But a new proof finds there are some objects whose differences are so stark, they can’t be effaced with a little more space. The work, posted at the end of May, solves a question posed by Charles Livingston in 1982 concerning…

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