Les Aventures d'Anselme Lanturlu - Le Geometricon by Jean-Pierre Petit

By Jean-Pierre Petit

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So {f, *} + {*, g} = {f+*, *+g) = {*+f, *+g) = {f,g}. 4 it suffices to prove CA- ` B2 represents zero. It does since we may factor of = gi and fig through the contractible space CA as follows A CA g B, 1. 2) we call the transgression square. Two "extreme" cases arise : (i) fi is a fibre map and v is the embedding of the fibre. Here the ladder collapses and we obtain either from S*(v) or from S*(J3) the homotopy exact sequence of the fibre map f3, namely .. - IIn (A Bo) IIn (A, B1) 8e1'-1 --C2'-iJ IIn-1(A, Bo) IIn (A, B2) ...

Square and k1 is a fibration, so is j2. Proof. We shall content ourselves with a proof of (i). fi (a2) This homotopy respects the relations since 9tj1(ao) =fij2 (ao). Clearly ftk1 = g t and ft is a homotopy of fo. In the light of this theorem we describe k1 as the cofibration induced by j2 and the map jl : A0 -+ A 1. Dually we describe j2 as the fibration induced by k1 and the map k2 : A2 -+ A3. Examples. If A2 =CA 0 andj2 is the embedding, then A3= u 1 ,CA0 INDUCED FIBRE AND COFIBRE MAPS 35 and k1 embeds A 1 in A3 in the obvious way.

Thus for inclusion maps we retrieve the relative homology groups and the usual exact homology sequence. We may show that H*(f) depends only on the homotopy class off. For if F : X x 1-+ Y is a homotopy from fo to fl, we have the diagram fo Y X "o iO 1 F XxI -'Y X Y where 0o and 01 are the indicated maps of'pairs. By the naturality of the exact sequence and the "five" lemma we infer that (Do*: H* (f0) H* (F) and (1 1 * : H* (ti) H* (F) proving that the homology groups of fo and fl are isomorphic.

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