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MATHEMATICAL TRIPOS Part III
Monday 6 June, 2005 9 to 12
PAPER 13
ALGEBRAIC TOPOLOGY
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.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Cover sheet None Treasury Tag Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
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
