PLUMBED HOMOLOGY oriented 3-manifold M 3 - PDF document

PLUMBED HOMOLOGY oriented 3-manifold M 3 n ~ 5 , C 3 . one can 3 3 M 3 M 3 The map C 3 • • (S3,K)] - [e(p,p-q)] is well defined though possibly not p/q A could yield new a homomorphism. Thus information about ~3 knowledge of C 3 ~ ~ M 3 ( = = )and is an oriented diffeomorphism type invariant. (ii). ~(M 3) (mod |6) = ~(M 3) (iii) For all known (Ma~ |979) ~/2-homolo y bordi of t ................ ~ sms such ~/2-spheres, ~ is invariant. If ~ is in fact a YL/2-homology bordism invariant and if it can be defined for any ~/2-sphere M 3 , triangulation problem plumbed 4-manifolds. i = ,s , S 2 2 ]). We call M(~) = ~P(~) the oriented 3-manifold obtained by ~lumbing disconnected plumbing = ~l + ) ~ = D 4 and M(@) = S 3 It is = e. i = j 13 I = l = 0 plumbing graph then M(P) that the for any j = ,s : lj 3J - Z only depends usual ~-invariant ~(M(P)) Proof. Everything except oriented diffeomorphism main ingredient however, since M 3 7Z/2-sphere and M 3 = W 4 oriented. Recall W 4 d 6 represents intersection number): x ~ smoothly embedded sphere W . It is easily seen even torsion even multiples = 0 . as in unique subset SC Vert(P) such d = where ~], natural basis H2(P(C);~) The defining property theorem. Condition adjacent vertices S . and d.d ~2R(M(p)) modulo ~(M(p)) only depends ~ ] F 2 above, then 0 e 2 • ~ _ ~ : 3 o I + down, and ~ , moves. This This , theorem 6.2] for the are 7£/2-spheres are ~ , one does quite general method generating homology spheres which homology null-bordant following simple V 4 connected oriented with connected boundary N 3 = A s , = O L.et M 3 be an A-sphere obtained N . ~.: SIX D 2 ) N , i = performed. Then ~ S 1 i = I, clearly represent = A s , exact homol- W 4 = M 3 . clearly A-acyclic, proving V 4 D 3 that any M 3 which results single index ~ . Mazur manifolds. k c c I c - - 6 2 b 2 b t b 2 b I 0 1 a 2 Then M(~l) results single index vice versa. Thus S 2 Mazur manifold plumbed ~/2-sphere = O . it is it is S ] K S 2 : [ ~ ~ that this so if isolated vertex O . it is [ ~ ] [ 6 [ ~ ] ) , i = I I = x I - - _ # 0 i . S l x S 2 i , b - = 0 [ 2 3 check, each family. The case. One c = r = l , , +cl,a ..... a s S ~ with patience. k � 0 . k ~ 0 reversing orientations k = 0 suitably interpreting First note (-l)-vertices directly -2 c-l e l which is equivalent, of 2.2, to the graph -2 c-1 c I represented in Kirby's link calculus [4-] , L eventually obtain (a|+k)-framed handle on the c| , next link by the argument which ~ is ~_,,"~,..._rJ, } 1 a 2 c 2 It is M 3 as in one has wE H2(P(F);~/2) some spin for all The Wu class basis ~1, linear combination well defined subset We call the sPi y . A subset Sc Sc ... ,s~ is the Wu set for some spin structure if it satisfies condition theorem 2.1, we call subset simply S~ J proof completely parallel ~(M(P),~) onlz depends oriented spin of (M0~),~) usual ~-invariant manifold (M(F),~). This involves strengthening proposition without spin structure [ ~ ] . shall need simple lemmas bilinear spaces, finite dimensional vector spaces equipped with bilinear ~ n symmetric bilinear space over 1 , V 1 only if sign V = non-degenerate symmetric bilinear space over I , V 1 V 1 Wu class unique element with w.x V 1 erate means V 1 ~ CVl where orthogonal complement. Now Now for some x . Since w.x = 0 have wE (Vt)m = V 1 V 1 w = I) ~ V = V 1 ~ V 1 V I since w(V~) # O . and we 2) = S 2 V I adding 2-handles D 4 S 3 = I = of 3.2 V 2 2 = S 2 V 2 4.2, that one can use S C $2C Vert(~ S C k . sees that Wu sets S 2 j , 2) - = $I ~ S 2 S 2 O , S 2 ~ = O . S I 2 Seifert spheres. from [~ ] . coprime integers, each ~. ~ 2 Then there exists Seifert manifold whose unnormalized Seifert invariants are (0;(~I'~I) .... '(~n'~n a~n(~-~'~i/a)= l manifold will denoted Z(~I ..... ~n every Seifert manifold which manifold after reversing orientation necessary. Various other descriptions these manifolds [ ~ ] [ @ ] , certain complex surface singularities. ~(~I .... ..... ~) ..... %) permit ~. = ] closed formula ~(~(~I ..... [ ~ ,theorem 6.~ seems fastest way only general formulae mod 2~ ° ~ 2C~'l " ..... %)) l . . . ) 1 [ ~ ] [ ~ ~ = ~ , , ] , , ] , ~" for ..... ~n-l'~n -b~ '''-~ .... k = [~I "'" ] + 2 2 , 4 . 8 , ~ 2~¢I - ~'n , like case above, then with minus = I . = O Until recently, almost ~ a ~/2-homology bordism M 3 = 0 ? uniform construction that the nontrivial obstruction seemed certain = M --~ -~I(W) probable restricted homology bordism homology bordism homology bordism M 3 , other sought One was connected 4-manifold if it M = V ? = O ? ~ . ~/2-spheres, keeping satisfying ~(-M) ~ , below holding T 3 = 8 the one M = ) - 4 + A , a 1 = a 3 = 1 a 2 = a 4 = 3 Browder Livesay yes for ~ , M 3 3 spin structure ~ on M with ~(M,~) nontrivial, (M,~) admits no orientation reversing # 0 , ~(-(M,~)) =-~(M,~) Siebenmann announced it is for all W 4 W 4 = O . 2 = M M . W = W l - W 2 = ~(M,~) -~(M,~) = O ~ = [ ~ ~ a n = 3 and low then also ~ 04 . n ! V , ~ ~ P(f") corresponding manifolds • " be the (-2)-weighted vertices k = 2 ~ , vertices correspond- vertex weighted S , V , whole chain (-2)-weighted exceptional would contradict negative definiteness V m ~(Im) =~(E)-mm smoothing singularities. ~ = G = C 3 f: C3---~ C W 4 = C 3 3 small) satisfies H| (W) = G , ) = O , 4 = M 3 2 , = 0 . V 4 homotopy equivalent W = = one can verify that M 3 3 . It are not 1 . which bound S 3 3(1967), 308-333, M ~ = 0