4D Crosscap Number

Every knot bounds a once punctured connected sum of real projective planes, RP2, in B4. The minumum number required is the 4D crosscap number. A knot is slice if and only if this number is 0. A tool for ruling out 4d-crosscap number ≤ 1 is a result of Yasuhara: If K bounds a punctured Mobius band in B4, then 4Arf(K) - signature(K) = 0, 2, or -2 mod 8. This applies in both the smooth and topological categories.


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