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The questions for the algebraic topology exam for part iii of the mathematical tripos, held on june 6, 2005. The exam consists of three questions, chosen from a total of five. The questions cover topics such as reduced homology groups, excision theorem, cup-product of cochains, compactly supported cohomology, thom isomorphism theorem, homotopy groups, and eilenberg-maclane spaces. The exam requires knowledge of general algebraic facts about short and long exact sequences, cell complex structures, and eilenberg-maclane spaces. The stationery requirements include a cover sheet and treasury tag, and no use of electronic devices is allowed.
Typology: Exams
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Monday 6 June, 2005 9 to 12
Attempt THREE questions.
There are FIVE questios in total. The questions carry equal weight.
Throughout, you may assume (i) general algebraic facts about short and long exact sequences (ii) closed manifolds admit cell complex structures which make a chosen closed submanifold a subcomplex (iii) the computation of the (co)homology groups of spheres (iv) Eilenberg-Maclane spaces are unique up to homotopy equivalence.
Cover sheet None Treasury Tag Script paper
1 What are the reduced homology groups H˜∗(X; Z) of a topological space X, and what are the relative homology groups H∗(X, A; Z) where A ⊂ X is a subspace?
Give a careful statement of the Excision Theorem. Deduce H∗(X, A; Z) ∼= H˜∗(X/A; Z) if X is a cell complex and A ⊂ X is a subcomplex. Hence, or otherwise, show that if Σg is a fixed closed oriented surface of genus g and A ⊂ Σg is an embedded circle then H∗(Σg , A; Z) determines whether A is nullhomologous but not whether it is nullhomotopic.
If A is an embedded disjoint union of 2 circles, what are the possibilities for H∗(Σg , A; Z)? Briefly justify your answer.
2 Define the cup-product of cochains, and prove that this product descends to a well- defined product in cohomology.
Describe without proof the cohomology ring of real projective space with Z 2 coefficients. Deduce that if n > m then f : RPn^ → RPm^ induces the zero map in reduced cohomology.
By considering the expression (^) |ff^ ((xx))−−ff^ ((−−xx))| , or otherwise, show that given any
f : Sn^ → Rn^ there is a point x ∈ S such that f (x) = f (−x). Hence deduce that if Sn^ is written as a union of (n + 1) closed sets, at least one of the closed sets contains a pair of antipodal points.
Would the last fact remain true if we took 4 closed sets on S^2? Justify your answer.
3 Define the compactly supported cohomology H ct∗(X; Z) of a space X, and compute H ct∗(Rn; Z).
State a version of the Poincar´e duality theorem for an oriented manifold M of dimension n. By considering the complement of a ball in M , or otherwise, show that if M is closed and oriented there is a map f : M → Sn^ of degree 1. Is there always a map f : Sn^ → M of degree 1? Justify your answer.
Finally, giving careful statements of any other standard results you use, show that the fibre bundle S^2 → CP^3 → S^4 (which you may assume exists) does not admit a section.
Paper 13