P R homomorphism P0 X : h0(X ) )h0(P) = R the is removed by tubular neighborhood >h,(Y).

11 o D(V) - S(V]. V (D(V),S(V)) J~ h*(V) and ) denote the resulting open embedding of exact sequence of the pair be the in Let D(V). j Then the becomes ~h* (D(V)) h* (S (V)) Replace h*(V) and replace by h*(X) h*(D(V)) by using the Thom isomorphism h*(X) using the isomorphism ~V sD(V). 12 ~ ' ,,,~h~ (X) IS (V)~D (V) h*(S(V)) where ¢ = sD(V) j , is multiplication by X_I(V). x for I. (l-t I) , x e h*(X). 12. 2. (D(V),S(V))~(D'(V@W),Sv(V@W)] homotopy equivalence of pairs and (SIxFD'(VOW), SIxFSv(VSW)) = (H,H~X) C(Z,X) homotopy equivalence and h*(Z,X) -- h*(sl×rD(V),SI×rs(v)).

On the c a t e g o r y A(R). Following [3], we extend C(G) LC(G) to will o r some c l o s e is a c o n t r a v a r i a n t in h* its one point at infinity. the c o h o m o l o g y as follows: If compacti~cation The action of X e LC(G), with G theory on + let denoting X+ is d e f i n e d by (i) (ii) Note that gx = X+(g)(x) +CX + set is a G invariant h*(X,Y) YCX ~ is a closed then R 9 + = +. algebra subspace. With f-l(K) f that is proper. is a compact subspace, so we define If this Y is the empty definition = h*(X÷,÷) w i t h unit Now suppose and that x ~ XCX + = h*(X+,Y +) h*(X) is an if g+ = +.

### A calculus for branched spines of 3-manifolds by Francesco Costantino

by Christopher

4.1