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  • Exercises #6 – Climbing Algebraic Topology – #4

    As a reference, we permit the use of homotopy theorems established in Topology and Geometry [Bredon1993] up till The Whitehead theorem. Exercise 1.Solution.Exercise 1. Let \(n\geq 2\) and let \(X\) be a homology \(n\)-sphere. If \(X\) is a simply-connected CW-complex, show that \(X\simeq \S^n\). Solution. Since \(X\) is simply-connected, by inverse Hurewicz, there is an…
  • Exercises #6 – Climbing Algebraic Topology – #3

    We use some of the results established in Topology and Geometry [Bredon1993] implicitly. In particular, the fact that the join of two spheres results in another sphere with degree \(n+m+1\) (which can also be proven independently using CW-structure) and some results about Stiefel manifolds. Exercise 1.Solution.Exercise 1. Show that \(X\) is Hausdorff if and only…
  • Exercises #5 – Augmentation Ideal

    Definition. For any group [latex]G[/latex], the augmentation map of the group ring [latex]\mathbb{Z} G[/latex] is defined to be the canonical ring homomorphism defined by [latex]\varepsilon(g)=1[/latex]. Exercise 1.Solution.Exercise 1. Prove the elements [latex]\{g-e: g\neq e\}[/latex] form a [latex]\mathbb{Z}[/latex]-basis for [latex]\ker \varepsilon[/latex]. Solution. Let [latex]I[/latex] be the free [latex]\mathbb{Z}[/latex]-module generated by [latex]\{g-e: g\neq e\}[/latex]. Note [latex]\varepsilon(g-e)=\varepsilon(g)-1=0[/latex], so…
  • Exercises #4 – Equalitarian Theories

    These were pulled from Exercises for [latex]\S5[/latex] of Chapter I in Bourbaki’s set theory book. (Assume [latex]\mathfrak{T}[/latex] has no constants, i.e. no letters in explicit axioms; thus we can abuse I.2.3.C3 without repercussions) Exercise 1.Solution.Exercise 1. Show the relation [latex]x=y[/latex] is functional in [latex]x[/latex] in [latex]\mathfrak{T}[/latex]. Solution. Denote the relation [latex]x=y[/latex] by [latex]R[/latex]. Let [latex]a[/latex] and…
  • Exercises #3 – Quantified Theories

    These were pulled from Exercises for [latex]\S[/latex]3 and Exercises for [latex]\S[/latex]4 of Chapter I in Bourbaki's set theory book. Exercise 1.Solution.Exercise 1. Let [latex]A[/latex] be a relation in logical theory [latex]\mathfrak{T}[/latex]. If [latex]A\iff (\mbox{not } A)[/latex] is a theorem in [latex]\mathfrak{T}[/latex], show that [latex]\mathfrak{T}[/latex] is contradictory. Solution. The negation of [latex]A\iff (\mbox{not } A)[/latex] is…
  • Exercises #2 – Independence of Axioms

    These were pulled from Exercises for [latex]\S[/latex]2 of Chapter I in Bourbaki's set theory book. Exercise 1.Solution for a.Solution for b.Exercise 1. Let [latex]\mathfrak{T}[/latex] be a theory, let [latex]A_1,A_2,...,A_n[/latex] be its explicit axioms and [latex]a_1,a_2,...,a_h[/latex] its constants. Let [latex]\mathfrak{T}'[/latex] be the theory with signs and schemes are the same as those of [latex]\mathfrak{T}[/latex], and whose…
  • Exercises #1 – Signs and Assemblies

    These were pulled from Exercises for [latex]\S[/latex]1 of Chapter I of Bourbaki's set theory book. Exercise 1.Solution.Exercise 1. Let [latex]\mathfrak{T}[/latex] be a theory with no specific signs. Show that no assembly in [latex]\mathfrak{T}[/latex] is a relation and that the only assemblies in [latex]\mathfrak{T}[/latex] which are terms are assemblies consisting of single letters. Solution. Let [latex]A[/latex]…