For a closed 4-manifold X and a knot K in the boundary of punctured X, we define gamma(0)(X)(K) to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured X with boundary K. Note that gamma(0)(S4) is equal to the non-orientable 4-ball genus and hence gamma(0)(X) is a generalization of the non-orientable 4-ball genus. While it is very likely that for given X, gamma(0)(X) has no upper bound, it is difficult to show it. In fact, even in the case of gamma(0)(S4), its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold X, has no upper bound. (C) 2015 Elsevier B.V. All rights reserved.
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