Realization Calculus: twin edition

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1 \chapter*{Notation and principal results}
2 \addcontentsline{toc}{chapter}{Notation and principal results}
3 \markboth{Notation and principal results}{}
4 All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by
5 \[
6 p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
7 \]
8 \begin{longtable}{@{}p{0.23\textwidth}p{0.71\textwidth}@{}}
9 \toprule
10 Symbol & Meaning\\\midrule
11 $C_p=\F_p$ & The additive cyclic group of prime order $p$.\\
12 $D=\bigoplus_p C_p$ & Finite-support prime-coordinate vectors.\\
13 $P=\prod_p C_p$ & All prime-coordinate vectors, with product topology.\\
14 $q:P\to V$ & The quotient map to $V=P/D$.\\
15 $D_B$ & The subgroup supported on prime indices in $B\subseteq\N$.\\
16 $H_W=q^{-1}(W)$ & The intermediate group associated to a rational subspace $W\le V$.\\
17 $S_A$ & The prime set $\{p_{2n+\one_A(n)}:n\in\N\}$.\\
18 $u_A$ & The coordinate indicator of $S_A$ in $P$.\\
19 $G_A$ & The group $q^{-1}(\Q q(u_A))$.\\
20 $s_A(r)$ & The specified rational lift of $r q(u_A)$ to $P$.\\
21 $c_A(r,t)$ & The finite cocycle $s_A(r)+s_A(t)-s_A(r+t)$.\\
22 $e_p$ & The vector supported at $p$ with value one.\\
23 $p\mid_G g$ & There exists $y\in G$ with $py=g$.\\
24 $A\eqfin B$ & The symmetric difference $A\triangle B$ is finite.\\
25 $\rad(m)$ & The product of the distinct prime divisors of $m$, with $\rad(1)=1$.\\
26 $\widehat G$ & The profinite completion, not the Pontryagin dual.\\
27 $A\le_T X$ & Membership in $A$ is computable using oracle $X$.\\
28 $K_{\mathfrak G}(G)$ & Least admissible finite description length, or $+\infty$.\\
29 \bottomrule
30 \end{longtable}
31 
32 \noindent The main statements have the following form:
33 \begin{align*}
34 \Tor(G_A)&=D,&G_A/D&\cong\Q,\\
35 G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\
36 G_A\cong G_B&\iff A\eqfin B,&&\\
37 G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
38 \end{align*}
39 The extension by $D$ does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.

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