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Diffeomorphisms of smooth manifolds. In this section we prove Theorem 111: Let C be a closed subset of a manifold M such that M - C is nonvoid and connected. Let a, b be arbitrary points of M - C. Then there is a diffeomorphism y : M -+ M homotopic to c M and such that ~ ( a= ) b and y ( x ) = x, x E C. T o this end we give the following lemma and its consequence. 36 I. Basic Concepts Lemma VI: There is a smooth functionf on R such that ( I ) carr f C [-3, 31 (2) 0 < f ( t ) < 1, ~ E andf(0) R (3) If'(t)l < 1, t E 08.

N Ok.. H e nc e p n 1 and s o p q n 1. < + + < + Case 11: x 4 U’. Then for each ki there is an element WE,C Wi:) such that x E W, . Moreover, the WE, are necessarily distinct. T h u s n Ws, n We,n ... , x is in p q distinct elements of W . It follows that p q 1~ 1. Distinguishing between the same two cases and using the fact that 9, and W are locally finite, we see that % ( l ) is locally finite. D. Lemma 111: If a manifold M has a basis 0, such that for each a, p, then for every open subset 0 of M dim 0, < dim 0

The space H ( X ) = Z ( X ) / B ( X is ) called the cohomology space of X . A homomorphism of differential spaces q ~ :(X, 6,) -P (Y, 6,) is a linear map for which 'p 6, = 6,o 9. It restricts to maps between the 0 cocycle and coboundary spaces, and so induces a linear map px : H ( X ) + H ( Y ) . A homotopy operator for two such homomorphisms, q ~ $, , is a linear map h: X --+ Y such that p- =h 8 s 0 h. + If h exists then q ~ #= $# . 0 + 0. Algebraic and Analytic Preliminaries 10 Suppose O - X - - - t Yf - Z - O &?