SURGERY ON TREES ANDREW RANICKI • Homotop y vs. homeomo rphism • T ransversalit y • Co dimension 1 submanifolds • Seifert-V an Kamp en Theo rem • Ma y er-Vieto ris exact sequence • Algeb raic K - and L -theo ry 1
Homotop y vs. homeomo rphism • A class of manifolds is rigid if every homotop y equivalence of manifolds in the class is homotopic to a homeomo rphism. • Main example of a rigid class: hyp erb olic manifolds, including all o riented 2-dimensional manifolds. • Bo rel conjecture: aspherical manifolds a re rigid. • In general, manifolds a re not rigid and there a re many distinct homeomo rphism classes of manifolds within a homotop y t yp e. 2
Surgery obstruction theo ry • Surgery theo ry p rovides systematic obstruc- tion theo ry in dimensions n ≥ 5 fo r deciding if a homotop y equivalence of n -dimensional : M n → X manifolds f is homotopic to a homeomo rphism. Obstructions involve the algeb raic K - and • -theo ry of mo dules and quadratic fo rms L over the group ring Z [ π ] of the fundamental group π = π ( X ). 1 3
Co dimension 1 metho ds in top ology 2 ⊂ M 3 • Study of 3-manifolds via surfaces N . • The Eilenb erg{Steenro d excision axiom fo r homology is a co dimension 1 transversalit y requirement. 1 ⊂ M n submanifolds N n − • Co dimension 1 pla y a central role in: Novik ov's p ro of of the top ological in- { va riance of the rational P ontrjagin classes (1966) Kirb y-Sieb enmann structure theo ry of high { dimensional top ological manifolds (1969) Chapman's p ro of of the top ological { inva riance of Whitehead to rsion (1974) controlled top ology . { 4
Geometric transversalit y • Let X b e a space with a subspace Y × R ⊂ X . Identify Y = Y × { 0 } ⊂ Y × R . • T ransversalit y theo rem: every map from an n -dimensional manifold : M n → X f is homotopic to a map which is transverse at Y ⊂ X , with N n − 1 = f − 1 ) ⊂ M n ( Y a co dimension 1 submanifold. 5
Splitting homotop y equivalences : M n → X • A homotop y equivalence f splits along Y ⊂ X if it is homotopic to one fo r which the restrictions : N n − 1 = f − 1 f | ( Y ) → Y , 1 = f − f | : M \ N ( X \ Y ) → X \ Y a re also homotop y equivalences. • Co dimension 1 splitting necessa ry along all Y ⊂ X (and sometimes su�cient) fo r f to b e homotopic to a homeomo rphism. • W aldhausen (1969) p roved that Hak en 3- dimensional manifolds a re rigid, using co di- mension 1 splitting metho ds. 6
Non-splitting homotop y equivalences • In general, homotop y equivalences do not split along co dimension 1 submanifolds, with b oth K - and L -theo ry obstructions. • F a rrell and Hsiang (1970) p roved that fo r : M n → n ≥ 6 a homotop y equivalence f 1 1 X × S splits along X × { pt . } ⊂ X × S if and only if τ ( f ) ∈ im (Wh( π ) → Wh( π × Z )), π = π ( X ) (Whitehead to rsion). 1 • Capp ell (1972) used a pa rtial computation of the L -theo ry of the in�nite dihedral group D ∞ = Z 2 ∗ Z to construct homotop y equiv- 2 +1 → RP 4 k 4 k +1 4 k +1 alences h : M # RP fo r k ≥ 1 which do not split along the sep- a rating co dimension 1 submanifold in the 4 k ⊂ RP +1 . 4 k +1 4 k connected sum S # RP 7
Object of the exercise Invent algeb ra which is su�ciently �exible • to have the geometric transversalit y p rop- erties of manifolds. • Identify the di�erence b et w een homotop y equivalences and homeomo rphisms of man- ifolds with the extent to which the K - and L -groups of the fundamental group ring [ π ] have this �exibilit y . Z • Computations in algeb raic K - and L -theo ry a re used in t w o directions, to p rove that: some homotop y equivalences of mani- { folds a re de�nitely homotopic to home- omo rphisms others a re de�nitely not homotopic to { homeomo rphisms. 8
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