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Embeddings and immersions of a $2$-sphere in $4$-manifolds
1993
Proceedings of the American Mathematical Society
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Let M be CP2#(-CP2)#P\#■■ -#Pm+k , where P,.Pm+k are copies of -CP2 . Let h, g, gx, ... , gm+k be the images of the standard generators of H2(CP2 ; Z), H2(-CP2 ; Z), H2(Pi; Z), ... , H2{Pm+k ; Z) in H2(M; Z) respectively. Let £ = ph + qg + J21Lx riSi he an element o\H2{M; Z). Suppose t,2 = I > 0 , p2 -q2 > 8 , \p\ -\q\ > 2 , and r, / 0 , i = 1, ... , m . If 2(m + I -2) > p2 -q2 , then £ cannot be represented by a smoothly embedded 2-sphere. If 2(m+r+[(l-r-l)/4]-1) >p2-q2 for some r with 0 < r <
doi:10.1090/s0002-9939-1993-1152976-1
fatcat:pkwxf67rpzfttkc7apltkje4ju