By Iain T. Adamson

This paintings goals to give uncomplicated topology in an unconventional method. It provides a evaluation of the fundamental definitions including workouts with no recommendations or proofs of the theorems partly 1, after which provides the ideas partially 2, permitting the scholar to check solutions with their very own.

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**Example text**

Sp+XS 2 1) , _~ , ( ~ ( S ~ _ × S 1 ) ) = ,(dpXO×S 1) (corresp. " , such that we have w ( ,^ ( x ' , y ) --~ Tc" = ^C-dp, ~ -TC" = (dpXD 2) % , ( ~ = = A~ . ,(= '-, ,(x , , x o x y , . ~ ~ - l ' c " { ~ , ~^'c -TC), = ~' r- 1 . ^{ c ' ) . XD 2,) % ^ ~ (A-TC), 47 where ~': d p X s I ~ -T-l(dp), are some trivializations of Let e': p- XS 1 ~d X s I p ~": - i ( dp ) (~p) > ~-I -and > dp > dP . > ~(d ×oxsl)(corresp. (A-TC), and because of the commutativity of the diagrams, S 2 XS I p- e' ~ #(d XOXS I) p pr[ S2 p- S2p+XS 1 I~_ ~' -- ~ d and e" > ~ ( 2 xOXS I) P pr i [~ S2 P+ p ~' > dA P A we can identify X 1 with ~ --TC" and X2 with A-TC'.

A surgery This immediately Y1' and ¥2' transforms X' into a connected sum of two copies of S 4, that is, into S 4. D. T h e o r e m 2. [E] Let W be a 3 - d i m e n s i c n a l be a complex (analytic) compact complex manifold, line bundle on W, ~O and ~i be two global (holomorphic) cross-sections (~i) ° of is a complex submanifold V 1 of W, the singular ~i of the zero-divisor (~o) ° = V o of locus S of all ordinary complex curve. following system of [E]. ~o singularities Suppose that the zero-divisor is canonical and the of V ° is an irreducible Suppose also that V 1 is transversal sense: V (Zl,Z2,Z3) hood U x of x in W, locus to V ° in the x E V 1 N V ° there exists a local complex coordinate on W with the center x such that in some neighbor- V 1 is defined by the equation z} = O and V ° is defined either by the equation z I = O or by the equation ZlZ 2 = O.

71,72 . 86 First of all we shall prove that there exists a diffeomorphism ~: X' > (S1XS3)l~ (SIxs3)2 such that for i = 1,2 (connected ~(yi ) = (slxai)i c (slxs3)i J al,a 2 E S 3. o Y'l = n(Yi )' i = 1,2, Let sum of copies of SIxs 5) o o q = Y1 A Y2 and B be a small There exist differential closed 2-disk in YI- p with center q. embeddings: ~i: I X I such that ~l(iXi) i = 1,2 > YI- p - int B, n ~2(ixI) = ~, ~i(zx~z) c ~B, ~i(ixI) = Yi-Yi~B, i = 1,2 (see Fig. 4). Denote A i = ~i(I×I), A = A I U A 2 U B.