~B(z) -id(R q) bundle) ZxR p ~ RP× Z) the ~ - s t r u c t u r e d of . 21), by the derivative map The two maps first bundle signs some fac- on the (-I) q+pr+pq+qr, L = Z × M X tubular by differ. (-I) (q+qr) (~B)*[g] = B*~*[g] be avoided: +I .

Is h o m o m o r p h i c , on inclusion maps we define - X] = [ g ' : M ~ X] if , A' c X' then > 0 J ( x ' , A ') t Rp . Similarly, : 0J(x,A) same . a n d it m a k e s of open subsets of j a contravariant 0e Rp . e. compact subsets X c Rp . iff they are locally closed iff they admit an open neighborhood is (relatively) = li B ~O~(V),V closed in V . For such is an open neighborhood The direct limit is taken over the direct V X in Rp we define of X ~ . system of groups oJv and @ homomorphisms ~ :oJv - 0iv ' .

Thus not depend ~ (in fact, o n l y o n the sense which we shall not > 0 (Y,B) is maps a : (Y,B) Moreover, on . it s h o w s t h a t : 0 (X,A) > (X,A) ~o ~ ~I ~ must X,Y is a s i n g l e G-manifolds again the map [N] | Any a general X = pt bordism Assuming Thus does such So* = ~I* = invariance). 17) E x a m p l e s . 0 (pt) ; moreover, of discuss ~*(a) ) [proj satisfy point N 0~X . For . 16. smooth we consider : Y × X × N <~,g>-1(0) > ~P <~,g>(y,x,n) = { ( y , x , n ) Ix = ~(y)} The trivialization of the normal = bundle = ~(y) graph(a) of - x × N ~ Y × N <~,g>-1(O) .

### Algebraic Topology: Proceedings, University of British Columbia, Vancouver, August 1977 by Roy Douglas (auth.), Peter Hoffman, Renzo A. Piccinini, Denis Sjerve (eds.)

by George

4.3