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The mathematical concepts of complex cobordism theory, focusing on complex structures on continuous maps, gysin maps, exotic characteristic classes, and the quaternionic projective plane. It includes explanations, proofs, and computations. Students of advanced mathematics, particularly those specializing in topology or algebraic geometry, will find this document useful for understanding complex cobordism theory.
Typology: Exams
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Thursday 7 June 2007 1.30 to 4.
Attempt FOUR questions.
There are FIVE questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Let Ω∗ U (·) be complex cobordism, i.e. the cobordism theory corresponding to stable complex structures in vector bundles. Explain what is meant by a complex structure on a continuous map f : X → Y of C∞-smooth manifolds, and for a fixed complex structure on f define (without proofs) the Gysin map f! in Ω∗ U (·). Prove that if L is a submanifold of a manifold M and the normal bundle ν of the embedding i : L ⊂ M has a stable complex structure, then i∗i!(1) ∈ Ω^2 Un (L)
is the top Chern class of ν in complex cobordism (dimC ν = n). You may assume that all manifolds in this question are compact and without a boundary.
2 Let η be a vector bundle, dimR η = n, over a smooth base space X and with a framed structure, i.e. with a continuous choice of an (ordered) orthonormal frame in each fiber of η. By considering the appropriate (non-ordered) Stiefelization of η, for each k = 1, 2 ,... , n construct an
( (^) n k
-sheeted cover pk : Xk → X, and define exotic characteristic classes of η by lk(η) = (pk)! (1) ∈ Ω^0 f r (X), k = 1,... , n
Deduce the Whitney sum formula for lk:
lk(η ⊕ ζ) =
i+j=k
li(η)lj (ζ),
for two framed bundles η and ζ.
3 Define the d 1 -metric on the space of C∞-maps of a compact manifold M ⊂ Rk into the Euclidean space RN^. Let f 1 , f 2 , ... ∈ C∞(M, RN^ ) be a sequence of maps which converges with respect to the d 1 -metric to an embedding i : M ⊂ RN^. Prove that there is an N 0 such that for any n > N 0 the map fn : M → RN^ is an immersion. Assuming that the second derivatives of all the fn, n = 1, 2 ,... are bounded by a constant C, prove that there is an N ′^ such that for any n > N ′^ the map fn : M → RN^ is an embedding.
Paper 19
