By Andrew H. Wallace

This self-contained remedy assumes just some wisdom of genuine numbers and genuine research. the 1st 3 chapters specialise in the fundamentals of point-set topology, and then the textual content proceeds to homology teams and non-stop mapping, barycentric subdivision, and simplicial complexes. routines shape a vital part of the textual content. 1961 version.

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C t) > 0 It1 > C (~(I~)) > C (i(zi)) > 0, denotes the c o c h i n complex of a graded Lie algebra. the isomorphism M(A~) ~ c (L(Z~)) which i s only formal, the cochain d i f f ~ e n t i a l terms of ~ DG-algebra~ pres~ves differentials. dc AM A~ For describes only the quadrate dM. The l-cochains u(i/j ) e M(I~), dual to the basis elements described earlier, are mapped to the cocycles relation in M(I£) z(i/j ) E Z£ c A%. 23) dMU(i/j ) = cic J shows that ~ coformal for has non-quadratic ~ > i. terms (for cj decomposable); The minimal algebras of the form hence C (L(Z%)) I~ is not have the homotopy type of a finite wedge of spheres, and their study goes back to P.

L(ii). Having now expressed our problem in terms of bordism theory we can translate it into homotopy theory. R. Wells [26] has shown that replacing embeddings by immersions in bordism theory corresponds to replacing homotopy groups of Thom complexes by their stable counterparts. e. is isomorphic to Sn+I(SI ) m Sn, the stable n-stem M0(1)) (S 1 = MS0(1)). 28 So the geometrical diagram SI (n,l) ~ e forget I orientatzon~ 6 ~ ~2 l ( n , l ) / / translates into homotopy theory as S ~ ~Sn+I(SI ) ~ n 2~2 ~+I (P~) 11 0 To study the problem using this diagram we must identify theoretically.

AMS 72 (1966), 358-428. groups for ~xG. 90 (1969), 226-234. Classification problems in topology IV (thickenings) Topology 5 (1966), 73-94. C. University Surgery on compact manifolds. of Warwick Coventry CV4 7AL, England Academic Press 1970. University of Geneva, Switzerland (current address) Homotopy invariants of foliations by S. Hurder and F. W. Kamber *) i. >.. A'(X), where I(G)£ ~, model [S]. G-structure on MA denote Let > 2£. DG-algebra A, The index £ depends on (X) = ~*(A'(X)) Sullivan-Dupont.