ON BROWDERS 1962 THEOREM ON THE HOMOTOPY TYPES OF DIFFERENTIABLE - PowerPoint PPT Presentation

1 ON BROWDER’S 1962 THEOREM ON THE HOMOTOPY TYPES OF DIFFERENTIABLE MANIFOLDS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar Edinburgh, 17 May, 2012

2 Finite H -spaces are Poincar´ e duality spaces ◮ An n -dimensional Poincar´ e duality space X is a space with a fundamental class [ X ] ∈ H n ( X ) and Poincar´ e duality isomorphisms [ X ] ∩ − : H ∗ ( X ) ∼ = H n −∗ ( X ) . ◮ An oriented n -dimensional manifold M is an n -dimensional Poincar´ e duality space, as is any space X homotopy equivalent to M . ◮ Theorem (Browder, 1961) If X is a path-connected H -space with f.g. (finitely generated) homology groups H ∗ ( X ) then X is an n -dimensional Poincar´ e duality space, with n = max { i | H i ( X ) ̸ = { 0 }} . ◮ Proved in Torsion in H -spaces , Annals of Maths. 74, 24–51 (1961) using heavy duty homological algebra. ◮ Question Is every path-connected H -space with f.g. homology homotopy equivalent to a manifold, e.g. a compact Lie group?

3 The surgery classification of exotic spheres ◮ Kervaire and Milnor obtained the surgery classification of exotic spheres in Groups of homotopy spheres I. , Annals of Maths. 77, 504–537 (1963) ◮ If M is a framed n -dimensional manifold with ∂ M a homotopy ( n − 1)-sphere there is defined a normal map (an anachronism) f : ( M , ∂ M ) → ( D n , S n − 1 ) with ∂ f a homotopy equivalence. ◮ K+M proved that for n � 5 there exist surgeries on the interior of M resulting in a framed cobordant contractible manifold M ′ if and only if the surgery obstruction σ ( f ) ∈ P n is 0. ◮ For n = 4 k σ ( f ) = signature( H 2 k ( M ; Q ) , b ) / 8 ∈ P 4 k = Z ◮ For n = 4 k + 2 σ ( f ) = Kervaire-Arf( H 2 k +1 ( M ; Z 2 ) , q ) ∈ P 4 k +2 = Z 2 ◮ For n = 2 j + 1 P 2 j +1 = 0, no obstruction.

4 Browder’s and Novikov’s questions ◮ In dimensions n = 1 , 2 every n -dimensional Poincar´ e duality space is homotopy equivalent to an n -dimensional manifold, and every homotopy equivalence of n -dimensional manifolds is homotopic to a diffeomorphism. ◮ Browder’s manifold existence question When is an n -dimensional Poincar´ e duality space X homotopy equivalent to an n -dimensional manifold? ◮ Novikov’s manifold uniqueness question When is a homotopy equivalence h : M → N of n -dimensional manifolds homotopic to a diffeomorphism? ◮ In 1962 Browder and Novikov applied the Kervaire+Milnor surgery method to their questions, obtaining very nice answers in the simply-connected case with n > 4. ◮ Thus began the Browder-Novikov-Sullivan-Wall surgery theory , which also deals with the non-simply-connected case, but still subject to n > 4.

5 Browder’s 1962 theorem ◮ Article in the mimeographed 1962 Arhus conference proceedings. ◮ Only published much later: Homotopy type of differentiable manifolds Proc. Novikov conjecture conference I., LMS Lecture Notes 226, 97-100 (1995) ◮ Theorem (B., 1962) Let X be a finite polyhedron which is an n -dimensional Poincar´ e duality space. If X is simply-connected and n � 5 then X is homotopy equivalent to an n -dimensional manifold if and only if there exists a j -plane vector bundle ν over X such that the fundamental class [ X ] ∈ H n ( X ) ∼ = H n + j ( T ( ν )) is represented by a map ρ : S n + j → T ( ν ) transverse regular at X ⊂ T ( ν ), and such that the normal map (anachronism!) = ρ | : M = ρ − 1 ( X ) → X f has surgery obstruction σ ( f ) = 0 ∈ P n .

6 The simply-connected surgery obstruction ◮ The homology groups of M split as H ∗ ( M ) = K ∗ ( M ) ⊕ H ∗ ( X ) with K ∗ ( M ) = ker( f ∗ : H ∗ ( M ) → H ∗ ( X )). ◮ For n = 4 k σ ( f ) = signature( K 2 k ( M ; Q ) , b ) / 8 ∈ P 4 k = Z ◮ For n = 4 k + 2 σ ( f ) = Kervaire-Arf( K 2 k +1 ( M ; Z 2 ) , q ) ∈ P 4 k +2 = Z 2 ◮ For n = 2 j + 1 P 2 j +1 = 0, no obstruction.

7 Browder’s converse of the Hirzebruch signature theorem ◮ (H., 1954) For a 4 k -dimensional manifold M = signature( H 2 k ( M ; Q ) , b ) signature( M ) = ⟨L ( M ) , [ M ] ⟩ ∈ Z with L ( M ) = L ( τ M ) ∈ H 4 ∗ ( M ; Q ) the L -genus of the tangent bundle τ M . ◮ The simply-connected surgery obstruction of a 4 k -dimensional normal map f : M → X is σ ( f ) = (signature( M ) − signature( X )) / 8 = ( ⟨L ( − ν ) , [ X ] ⟩ − signature( X )) / 8 ∈ P 4 k = Z . ◮ (B. 1962) For k � 2 a 4 k -dimensional Poincar´ e duality space X is homotopy equivalent to a manifold if and only if there exists ν with signature( X ) = ⟨L ( − ν ) , [ X ] ⟩ ∈ Z .

8 Are finite H -spaces homotopy equivalent to manifolds? ◮ Some 40 years after Browder’s original 1961/2 question Bauer, Kitchloo, Notbohm and Pedersen finally proved that every finite H -space is in fact homotopy equivalent to a manifold! ◮ Finite loops are manifolds Acta Math. 192, 5–31 (2004) ◮ The proof used a combination of homotopy theory and non-simply-connected surgery theory.

9 Books on surgery theory ◮ C.T.C. Wall, Surgery on compact manifolds Academic Press (1971) and AMS (1999) ◮ W. Browder, Surgery on simply-connected manifolds Springer (1972) ◮ A. Ranicki, Algebraic L -theory and topological manifolds CUP (1992) ◮ A. Ranicki, Algebraic and geometric surgery OUP (2002)