M8130 Algebraic topology, tutorial 05, 2024_ 24. 10. 2024
Exercise 1. Prove that Sn has a nonzero vector field if and only if n is odd.
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M81S0 Algebraic topology, tutorial 05, 2024
24. 10. 2024
Exercise 2. Compute homology groups of oriented two dimensional surfaces using a suitable structure of CW- complex.
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M8130 Algebraic topology, tutorial 05, 2024_H- 10- 2024
Exercise 3. Have f:Sn^Sn map of degree k.  (such map always exists). Let X = Dn+1 Uf Sn and compute homology of X and the projection p: X -4 X/Sn in homology.
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M8130 Algebraic topology, tutorial 05, 2024_ 24. 10. 2024
Let X be a topological space with finitely generated homological groups and let H{(X) = 0 for each sufficiently large i. Every finitely generated abelian group can be written as Z @ Z © • • • © Z © Tor, where Tor denote torsion part of the group.   The number k is
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Euler characteristic x of X is defined by:
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{Z    i = 0 n Thus x(Sn) = 1- (-l)n. 0, otherwise.
Exercise 4. Let (C*,<9) 6e a c/iam complex with homology H*(C+). Prove that = X{C*), where
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M8130 Algebraic topology, tutorial 05, 2024
24. 10. 2024
Let X be a topological space with finitely generated homological groups and let Hi{X) = 0 for every sufficiently large i. Let f: X —> X be a continuous map. Map f induces ho-momorphism on the chain complex /*: C*(X) —f C*{X) and on the homologiy groups H*f: H*(X) H*(X), where HJ(TorH*(X)) C TovH*(X). Thus it induces homomor-phism
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Theorem. If L{f) 7^ 0, then f has a fixed point.
Exercise 5. Use the theorem above to show, that every continuous map f on Dn and RPn where n is even has a fixed point.
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M81S0 Algebraic topology, tutorial 05, 2024_24. 10. 2024
Exercise 6. Let M be a smooth compact manifold. Prove implication the => in the statement, that there is a nonzero vector field on M if and only if x{M) = 0.
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