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29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

19 Jan 2021: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

15 Oct 2021: b) Suppose that v V , with components vi and ω V with components ωj. ... Please send any corrections to c.warnick@warwick.ac.uk. 2. b) Fix x U and suppose that V ,W TxU are two vectors such that for any f

23 Nov 2021: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

15 Oct 2021: Exercise 3.1. a) Suppose that {j. }1j=1 E () is a sequence such that j! ... f 2 L1(Rn). Fix > 0. Show thatthere exists f. 2 C10(Rn) such that.

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... b) Fix a local base β̇. Show that a sequence (xn)n=1 is τ-Cauchy if and only if for anyB β̇ we can find an integer N such that.

26 Oct 2021: of B such thatπ1(Uα) is a disjoint union. ... H(B) H(E) such that π π! is multiplication by N.

16 Mar 2021: 7. Let R Q be a ring. If X is a 1-connected space such that H(X; R) = H(Sn; R),show that there is a map f : Sn X ... Hint: a) Show that such rings have Tor(,R) = 0;b) Consider the class of abelian groups A such that AR = 0.].

19 Jan 2021: i) Prove that fm is a group homomorphism. (ii) Let a,b Z be such that ab 1 modulo m. ... Deduce thatthere exists r > 0 such that. {n such that p | 2n 1} = rZ.