Perturbations Spectralsequences Cascades Multicomplexes Different Approaches to Morse-Bott Homology David Hurtubise with Augustin Banyaga PennStateAltoona math.aa.psu.edu Universit´e Cheikh Anta Diop de Dakar, Senegal May 19, 2012 DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Cascades Multicomplexes Computing homology using critical points and flow lines Perturbations Generic perturbations Applications of the perturbation approach A more explicit perturbation Spectral sequences Filtrations and Morse-Bott functions Filtrations and spectral sequences Applications of the spectral sequence approach Cascades Picture of a 3-cascade Applications of the cascade approach Cascades and perturbations Multicomplexes Definition and assembly Multicomplexes and spectral sequences The Morse-Bott-Smale multicomplex DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes The Morse-Smale-Witten chain complex Let f : M → R be a Morse-Smale function on a compact smooth Riemannian manifold M of dimension m < ∞, and assume that orientations for the unstable manifolds of f have been chosen. Let C (f) be the free abelian group generated by the critical points of k index k, and let m M C (f) = C (f). ∗ k k=0 Define a homomorphism ∂ : C (f) → C (f) by k k k−1 X ∂ (q) = n(q,p)p k p∈Cr (f) k−1 where n(q,p) is the number of gradient flow lines from q to p counted with sign. The pair (C (f),∂ ) is called the ∗ ∗ Morse-Smale-Witten chain complex of f. DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes The height function on the 2-sphere z 2 S n 1 f 0 ¡1 s C (f) ∂2 //C (f) ∂1 //C (f) //0 2OO 1OO 0OO ≈ ≈ ≈ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) < n > ∂2 //< 0 > ∂1 //< s > //0 DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes The height function on a deformed 2-sphere TsuM z s r +1 Eu S2 ¡1 q f +1 ¡1 p C (f) ∂2 //C (f) ∂1 //C (f) //0 2OO 1OO 0OO ≈ ≈ ≈ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) < r,s > ∂2 //< q > ∂1 //< p > //0 DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes References for Morse homology I Augustin Banyaga and David Hurtubise, Lectures on Morse homology, Kluwer Texts in the Mathematical Sciences 29, Kluwer Academic Publishers Group, 2004. I Andreas Floer, Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), no. 1, 207–221. I John Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965. I Matthias Schwarz, Morse homology, Progress in Mathematics 111, Birkh¨auser, 1993. I Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661–692. DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Can we construct a chain complex for this function? a spectral sequence? a multicomplex? Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes A Morse-Bott function on the 2-sphere n z2 S2 1 f B 0 2 B 0 1 s DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Perturbations Spectralsequences Morsehomology Cascades Morse-Botthomology Multicomplexes A Morse-Bott function on the 2-sphere n z2 S2 1 f B 0 2 B 0 1 s Can we construct a chain complex for this function? a spectral sequence? a multicomplex? DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Theorem Let M be a finite dimensional compact smooth manifold. The space of all Cr Morse functions on M is an open dense subspace of Cr(M,R) for any 2 ≤ r ≤ ∞ where Cr(M,R) denotes the space of all Cr functions on M with the Cr topology. Why not just perturb the Morse-Bott function f : M → R to a Morse function? Perturbations Genericperturbations Spectralsequences Applicationsoftheperturbationapproach Cascades Amoreexplicitperturbation Multicomplexes Generic perturbations Theorem (Morse 1932) Let M be a finite dimensional smooth manifold. Given any smooth function f : M → R and any ε > 0, there is a Morse function g : M → R such that sup{|f(x)−g(x)| | x ∈ M} < ε. DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology Why not just perturb the Morse-Bott function f : M → R to a Morse function? Perturbations Genericperturbations Spectralsequences Applicationsoftheperturbationapproach Cascades Amoreexplicitperturbation Multicomplexes Generic perturbations Theorem (Morse 1932) Let M be a finite dimensional smooth manifold. Given any smooth function f : M → R and any ε > 0, there is a Morse function g : M → R such that sup{|f(x)−g(x)| | x ∈ M} < ε. Theorem Let M be a finite dimensional compact smooth manifold. The space of all Cr Morse functions on M is an open dense subspace of Cr(M,R) for any 2 ≤ r ≤ ∞ where Cr(M,R) denotes the space of all Cr functions on M with the Cr topology. DavidHurtubisewithAugustinBanyaga DifferentApproachestoMorse-BottHomology