Last update: 02 March 2012
HW: Let $G$ be a (commutative?) topological group with a fundamental system of neighborhoods of 0 which are subgroups, $$0\subseteq \cdots \subseteq {G}_{n+1}\subseteq {G}_{n}\subseteq \cdots \subseteq {G}_{2}\subseteq {G}_{1}.$$ Show that the completion of $G$ $$\hat{G}\simeq \underleftarrow{\mathrm{lim}}\frac{G}{{G}_{n}}$$ by showing that a sequence in $G$ is coherent if and only if it is Cauchy.
HW: Let $S$ be a set and let $n\in {\mathbb{Z}}_{>0}.$
HW: Define the notion of a function $f:X\to Y$ as a subset of $X\times Y$.
Hint: 

Consider the graph of $f$ and write down precise conditions for a subset $S$ of $X\times Y$ to be the graph of a function.
$\square $ 
HW:
HW: Let $X$ and $Y$ be topological spaces. Let $a\in X$ and let $f:X\to Y$ be a function.
HW: Show that ${\mathbb{Z}}_{\left(0\right)}=\mathbb{Q}.$
HW: Expand $\mathrm{tan}t$ as a power series beginning with $t.$
HW: Do the exercises on the padic numbers page.
HW: Using the definition of the ring of fractions with denominators in $S$ from the lectures, prove that $=$ is an equivalence relation, that $$+:A\left[{S}^{1}\right]\times A\left[{S}^{1}\right]\to A\left[{S}^{1}\right]$$ is well defined, that $$\cdot :A\left[{S}^{1}\right]\times A\left[{S}^{1}\right]\to A\left[{S}^{1}\right]$$ is well defined, and $A\left[{S}^{1}\right]$ with $+$ and $\cdot $ is a ring.
HW: Show that $$\begin{array}{rrcl}{S}^{1}u:& {S}^{1}M:& \to & {S}^{1}N:\\ & \frac{m}{s}& \mapsto & \frac{u\left(m\right)}{s}\end{array}$$ is an $A\left[{S}^{1}\right]$module homomorphism.
HW: Show that ${S}^{1}$ is a functor (i.e. ${S}^{1}({u}_{1}\circ {u}_{2})={S}^{1}\left({u}_{1}\right)\circ {S}^{1}\left({u}_{2}\right)$).
HW: Let $f:V\to E$ be a linear transformation from a vector space $V$ into an algebra $E.$ Suppose that if $v\in V$ then $f{\left(v\right)}^{2}=0.$ Show that if ${v}_{1},{v}_{2}\in V$ then $f\left({v}_{1}\right)f\left({v}_{2}\right)=f\left({v}_{2}\right)f\left({v}_{1}\right).$
HW: Compute the dimension of ${S}^{k}\left(V\right).$
HW: Determine (make precise) the definition of $\mathrm{sgn}\left(\sigma \right),$ as used in the construction of $\Lambda \left(V\right).$
HW: Show that if $G$ is a cyclic group then $G\simeq \mathbb{Z}$ or $G\simeq \mathbb{Z}/r\mathbb{Z}$ for some $r\in {\mathbb{Z}}_{\ge 0}.$
HW: Show that the group ${G}_{r,r,2}$ can be presented by generators ${s}_{1},{s}_{2}$ with relations $${s}_{1}^{2}=1,\phantom{\rule{2em}{0ex}}{s}_{2}^{2}=1,\phantom{\rule{2em}{0ex}}{\displaystyle \underset{rfactors}{\underset{\u23df}{{s}_{1}{s}_{2}{s}_{1}\cdots}}}={\displaystyle \underset{rfactors}{\underset{\u23df}{{s}_{2}{s}_{1}{s}_{2}\cdots}}}$$