By Iain T. Adamson

This paintings goals to offer undemanding topology in an unconventional method. It provides a assessment of the fundamental definitions including routines with no suggestions or proofs of the theorems partially 1, after which offers the suggestions partly 2, permitting the scholar to match solutions with their very own.

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**Example text**

To embedding. NI to get See also Lemma ~p Let N. 1. 38 A, the a t t a c h i n g M ! = ~+N I (M = ~_NI). an element N I = M I x I U (n-2)-handles, an element over matrix where represents in 3-handles n > 5, and = x. 1. 3. incare Dualit~ Let X be a f i n i t e = ~I(X,*) , and c E Cn(X)@~. define c ~ w:~ :cq(x) tr:C If is i n f i n i t e , ~ n are u s i n g Definition exists X and c n (X) ÷ H X n-q is acts the X ; we give Definition image has [X] (X;R~) of is the m a p p i n g on. e H n with ([A2], chaini formula over (X;~ ~ , R tr(c) pg.

Y,X) Then A i-th row, column of X rel X. "elementary" ~w, where operations Ri,C i denote g ~ w, a ~H(P) (2) Ci~+ +-aCig (3) R i~* R i + xRj (4) Ci~* C i + Cjx A = CBD is the identity A: R i ~+ -+agRi can be transformed Y /~ over (i) (1) - (4), then This matrix is invertible = 0. on an invertible matrix A (with respect to the (Y,YI,X). Consider the following If where coming from the exact Suppose the matrix of for some choice of the respectively, is defined to be the matrix of the map sequence of the triple since with generators are the attaching maps.

Fr a handlebody M is a c o b o r d i s m H = TOP. H = DIFF, Kirby N be a c o b o r d i s m is the the from x D n-k, __if of the This [DI9] if and N d i m N _> 6 of e m b e d d i n s s Proof: trace by a d d i n g Let with the f:S k x D n - k ÷ M f(S k x 0). N = M x I •D f is c a l l e d and straightening f:S k x D n - k ÷ M x l, t h e n and H-manifold = (M-f(Int(S k x Dn-k))) ~ flS k x S n - k - l , H = DIFF. the with and N is N = M ~-/ D k+l, f0 where an e x t e n s i o n X is ~:N ÷ X Define of c o m m u t a t i v e Wk+l(¢) group of h o m o t o p y classes diagrams Sk Dk+l C M There to be the )X is a long exact .