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  • So when our surgeries stop, we have a sphere.

    Now we reverse the process. Three kinds of desurgery can happen.

    (i) We have two discs with arrows going round in opposite directions, and sew in a cylinder. This is the same as sewing on a handle (Figure 126).

    (ii) We have one disc, and sew in a Möbius band.

    (iii) We have two discs with arrows going round the same way: sewing a cylinder back is the same as sewing on a Klein bottle (Figure 127), which is equivalent to sewing on two Möbius strips (compare Figure 83). So we can convert the third kind of desurgery into two of the second kind.

    —Ian Stewart, 1975, Concepts of Modern Mathematics

    Evidently a nonce-use of Stewart's to indicate the reversal of the topological process of surgery on a surface (starting with a sphere and increasing its genus or number of crosscaps).

    August 12, 2008