By Gerald A. Anderson

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**Sample text**

2. over R. degree Furthermore, 1. Our p r o b l e m i) When is the 2) When is ¢:M ÷ X Question 1 is best lifting X ~ of these notes. Wall [HI9] let subcomplexes of there a handle ¢i:Ni Assume X ~ Xi Cr+l:Nr+l cobordant have to a h o m o l o g y

Generated by Wh(w;R) H C(Rw)" More generally, homomorphism, if w i. Let and ~roup of w is a be the sub

Generator (2) It is e a s y pondence with and let Then ~ is a S p i v a k since T (b), (C M) , :Hn+ k sends map %X (sn+k;~) + H (T(~);~) n+k to g e n e r a t o r . the to see that NIH(x;~) set of h o m o t o p y is in i-i classes of corres- lifts BH I X "~ ,> (BG) f where f lifts g0,g 1 denotes Then the denote , G:X E ÷ X fiber the c l a s s i f y i n g are h o m o t o p i c X x I ÷ (BG)p f i = 0,1 . Let P if t h e r e x I ÷ BH denote of E ÷ X the a c t i o n of the map fiber ~p/H on 52 ~/ , a n d two exists , so t h a t the p u l l b a c k is for GIX E .