SLIDE 1
ON BROWDERS 1962 THEOREM ON THE HOMOTOPY TYPES OF DIFFERENTIABLE - - PowerPoint PPT Presentation
ON BROWDERS 1962 THEOREM ON THE HOMOTOPY TYPES OF DIFFERENTIABLE - - PowerPoint PPT Presentation
1 ON BROWDERS 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
SLIDE 2
SLIDE 3
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) → (Dn, Sn−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 ) ∈ Pn is 0.
◮ For n = 4k
σ(f ) = signature(H2k(M; Q), b)/8 ∈ P4k = Z
◮ For n = 4k + 2
σ(f ) = Kervaire-Arf(H2k+1(M; Z2), q) ∈ P4k+2 = Z2
◮ For n = 2j + 1 P2j+1 = 0, no obstruction.
SLIDE 4
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.
SLIDE 5
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] ∈ Hn(X) ∼ = Hn+j(T(ν)) is represented by a map ρ : Sn+j → T(ν) transverse regular at X ⊂ T(ν), and such that the normal map (anachronism!) f = ρ| : M = ρ−1(X) → X has surgery obstruction σ(f ) = 0 ∈ Pn.
SLIDE 6
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 = 4k
σ(f ) = signature(K2k(M; Q), b)/8 ∈ P4k = Z
◮ For n = 4k + 2
σ(f ) = Kervaire-Arf(K2k+1(M; Z2), q) ∈ P4k+2 = Z2
◮ For n = 2j + 1 P2j+1 = 0, no obstruction.
SLIDE 7
7 Browder’s converse of the Hirzebruch signature theorem
◮ (H., 1954) For a 4k-dimensional manifold M
signature(M) = signature(H2k(M; Q), b) = ⟨L(M), [M]⟩ ∈ Z with L(M) = L(τM) ∈ H4∗(M; Q) the L-genus of the tangent bundle τM.
◮ The simply-connected surgery obstruction of a 4k-dimensional
normal map f : M → X is σ(f ) = (signature(M) − signature(X))/8 = (⟨L(−ν), [X]⟩ − signature(X))/8 ∈ P4k = Z .
◮ (B. 1962) For k 2 a 4k-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 .
SLIDE 8
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.
SLIDE 9