By Matthias Kreck (auth.)

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7: This subtraction of handles is a special case of a more general construction which is the inverse process of the a d d i t i o n of handles and makes the notation subtraction of handles more clear (compare [ 4 9 ] ). Con- 46 sider d i s j o i n t embeddings s o and s I of (Dk,s k ' l ) into (NF,Fi) meeting Fi t r a n s v e r s a l l y and remove d i s j o i n t open tubular neighborhoods from NF. I I F0 F1 l The boundary of the r e s u l t i n g manifolds is equal to F0 , ! ) F 1 . But i t f is not clear that F0 = -F I .

The rank of W is equal to rank ~k(N F'' F'i ) / 2 and "ITk(N' F~' F'i ) / Ji. (W) is torsion free. 1. 5 in the even-dimensional case. 4: We consider the following diagram of exact sequences of homology groups with rational coefficients: 0 Hk+i (NF '@NF) Hk(QNF) 0 ) Hk(F) • ) Hk(NF) ~ Ji ) Hk(NF,Fi) i > Hk_I(F) >Hk_I(NF) ~ 0 Hk(NF,aNF) The zeros in the row follow from the condition that Hk+I(NF,Fi) : Hk-I(NF,Fi+I) = {0~ and Hk_I(NF,Fi) = fO~ . The zero in the column follows from the observation that by Poincar~ duality the map Hk+I(NF) >Hk+I(N F,BNF) corresponds to the map Hk-I(NF, aNF) >Hk-I(NF) which factorizes through Hk'I(NF,Fi) = { 0 } .

Let NF be normal bundle and we can this element. The reason is that the sphere bounds an embedded disk in NF. 5. Then by a sequence of handle additiomwe can replace NF by an admissible manifold N~ with the same properties as NF and such that in addition the boundary operators a i : 1T k (NF,Fi) > ~'k_l(F) are zero for k ~ (n-l)/2. 0 b) Subtraction of handles. Let k ~ n/2 and s : Sk ~N F be a d i f f e r e n t i a b - le embedding into the n-dimensional admissible manifold NF. I f k = n/2 we suppose that the s e l f intersection number of t h i s embedding is zero.