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

These were pulled from Exercises for $\S5$ of Chapter I in Bourbaki’s set theory book. (Assume $\mathfrak{T}$ 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 $x=y$ is functional in $x$ in $\mathfrak{T}$. Solution. Denote the relation $x=y$ by $R$. Let $a$ and…
• ### Exercises #3 – Quantified Theories

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

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

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