Quasirationality and aspherical (pro-$p$)presentations

TL;DR

This paper investigates quasirational (pro-$p$) presentations, exploring their properties and differences from aspherical presentations, and discusses implications for Whitehead's conjecture, confirming the existence of a broad class of such presentations.

Contribution

It introduces and analyzes quasirational (pro-$p$) presentations, highlighting their distinctions from aspherical presentations and confirming the existence of a large class of these presentations.

Findings

Quasirational (pro-$p$) presentations are distinct from aspherical presentations.

The study sheds light on Whitehead's conjecture.

Confirms the existence of a proper class of quasirational presentations.

Abstract

Quasirational (pro-$p$)presentations are studied. The difference between aspherical and quasirational presentations sheds some light on the Whitehead's conjecture. We confirm expectations O.V. Melnikov on existence of a proper class of presentations.

Equations

$$1\to R\to F\stackrel{\pi }{\to }G\to 1$$

$$C_{1}Y\to C_{0}Y\to \mathbb{Z}\to 0.$$

$$0\to H_1(G,IG)\to R/[R,F]\to {\mathbb{Z}}_{(p)}^{|X|}\to (IG{)}_G\to 0,$$

$$0\to \overline{R}\to {\mathbb{Z}}_{(p)}{G}^{|X|}\to IG\to 0,$$

$$0\to IG\to {\mathbb{Z}}_{(p)}G\to {\mathbb{Z}}_{(p)}\to 0.$$

$$Q_8=⟨a,b|a^4=b^2,aba=b⟩$$

$${\mathcal{T}}_n=((\frac{R}{[R,R{\mathcal{M}}_n]}{)}_p^{\wedge },{\varphi }_n^{n+1}),{\mathcal{T}}_n^p=(\frac{R}{R^p[R,R{\mathcal{M}}_n]},{\tilde{\varphi }}_n^{n+1}),$$

$${\varphi }_n^{n+1}:(R/[R,R{\mathcal{M}}_{n+1}]{)}_p^{\wedge }\twoheadrightarrow (R/[R,R{\mathcal{M}}_n]{)}_p^{\wedge },$$

$${\tilde{\varphi }}_n^{n+1}:R/R^p[R,R{\mathcal{M}}_{n+1}]\twoheadrightarrow R/R^p[R,R{\mathcal{M}}_n],$$

$${\mathbb{Z}}_p[\zeta ]{\uparrow }_H^G:={\mathbb{Z}}_p[\zeta ]G{\otimes }_{{\mathbb{Z}}_p[\zeta ]H}{\mathbb{Z}}_p[\zeta ]$$

Full text

: 22 .

: , , .

00footnotetext:

1

[15] ( -$p$) , , -$p$- -$p$- .

• ( . [10, 11, 9]), ( -$p$) ( . . [3]) , -$p$- [15, 16, 17]. ‘‘ ’’ ( ), 2 $p$- ( , $Q_{8}$).

2 , [15, 1] ( -$p$) ( -$p$) $G$ , $H_{2}(G,\mathbb{Z})$ ( $H_{2}(G,\mathbb{Z}_{p})$ -$p$- ) . ( -$p$) ( -$p$) $G$ ( -$p$) ( ) . , $Q_{8}$ , -$p$- ( [3] , $p$- ).

3 * * ‘‘ ’’ [3] $\mathbb{Z}_{p}$ $\mathbb{F}_{p}$- , $CA$- ( $CA$- ) -$p$- ( [3] , $\mathbb{F}_{p}$- -$p$- $CA$- ). 1 $mod(p)$ $QR$- ( 3 ). , $QR$- . . , 2 (‘‘ ’’). , . . -$p$- ( -$p$- [4], [1]), 2 .

. . . . , , .

2

$G$ -

[TABLE]

$F=F(X)$ $X$, $R$ - $F$, $r\in R$. -$p$- , $p$- . ( ), . [2], [21]. $X$ - , -$p$- -$p$- $(F(X),i:X\hookrightarrow F(X)),$ . , $\phi:X\rightarrow G$ -$p$- $G$ , $X$ $G,$ -$p$- $\widetilde{\phi}:F(X)\rightarrow G,$ [21, 3.3]. $X$ - , -$p$- , $F(X)=\varprojlim_{U\triangleleft\Phi(X)}\Phi(X)/U,$ $\mid\Phi(X)/U\mid=p^{n}$ - -$p$- $\Phi(X)$ ( $X$) -$p$- $\Phi(X),$ $p$. , , -$p$- $G$ -$p$- , $G$ (2.1), $F$ - -$p$- ( $X$ - ), $R$ - , $F$. -$p$- -$p$- , $F$ $G$.

$C=\varprojlim C_{\alpha}$ - ($C_{\alpha}$ - ), $CG$ - -$p$- $G$. $CG=\varprojlim CG_{\mu}$ [21, 5.3], $G=\varprojlim G_{\mu}$ - -$p$- $G$ $p$- $G_{\mu}$.

, $p$ , -$p$- $p$ . $G$ - ( -$p$) ( -$p$) (2.1), $\overline{R}=R/[R,R]$ *$G$- *, $[R,R]$ - , $G$ $F$ $R$. $p\geq 2$ ${\Delta}_{p}$ - $\mathbb{F}_{p}G,$ $\mathcal{M}_{n},n\in\mathbb{N}$ $p$- $F$ $\mathbb{F}_{p}$, $\mathcal{M}_{n}=\{f\in F\mid f-1\in{\Delta}^{n}_{p}\}.$ -$p$- ${\Delta}^{n}$ $n$- $\Delta={\Delta}_{p}$, - $n$- ${\Delta}_{p}$ [20]. -$p$- [2, 7.4], [20, .11], .

1

[15, 1]** (2.1) ($QR$- ), $n>0$ $p\geq 2$ $F/R\mathcal{M}_{n}$- $R/[R,R\mathcal{M}_{n}]$ $p$- ($p$ -$p$- $p\geq 2$ $p$- ). .

( 1 1 2 ), 1 . , $p$- $R/[R,R\mathcal{M}_{n}]$ $p$- $\varprojlim(R/[R,R\mathcal{M}_{n}])^{\wedge}_{p}$ $\overline{R}\widehat{\otimes}\mathbb{Q}_{p}=\varprojlim_{n}R/[R,R\mathcal{M}_{n}]\otimes\mathbb{Q}_{p},$ $\mathbb{Q}_{p}$ - $p$- . -$p$- $\overline{R^{\wedge}_{w}}(\mathbb{Q}_{p})$ [17, 2], $p$- $R$. $\overline{R}\widehat{\otimes}\mathbb{Q}_{p}$ [17, 1 2] . , [18, 1] ‘‘ ’’ -$p$- c , [17, 3] 2 -$p$- .

1

[15, 4]** ( -$p$) (2.1), :

(i) (2.1) $QR$-( -$p$) ;

(ii) ( -$p$) - $R/[R,F]$ .

, ( -$p$) , $p,$ $\mathbb{F}_{p},$ , [15, 16, 17].

(2.1) - , $K(X;R)$ $CW$- ( [9]). (2.1) ** **, $K(X;R)$ , $\pi_{q}(K(X;R))=0$, $q\geq 2$.

, . , $K(X;R)$ $\widetilde{K(X;R)}$ [7, I, 5.3] $\mathbb{Z}G$- $Y$:

[TABLE]

$H_{1}Y\cong(\pi_{1}Y)_{ab}\cong\overline{R}$ $G$- $H_{1}Y.$

$r\in F$ - $F$, $\sqrt{r}$ $F$, - $s\in F,$ $s^{m}=r$ $m$ ( $F$ - , ). . (**CA- **),

2

[9, 1]** (2.1) $G$ $CA$- , $\overline{R}$ $\mathbb{Z}G$- ( ) $P_{r},r\in R$, $P_{r}$ $\overline{r}=r[R,R]$ $\pi(s)\overline{r}=\overline{r},s=\sqrt{r}$, $\pi$ - (2.1).

$CA$- , $P_{r},r\in R$ 2 $\mathbb{Z}G$, [8]. , $CA$- , ( ).

$QR$-( -$p$) , , $"$ $"$ [8], -$p$- , -$p$- [5, 10.2]. , $QR$– $CA$- [15, 3] ($R/[R,F]\cong\oplus_{r}(P_{r})_{F}\cong\oplus_{r}\mathbb{Z}$ $F$ ), -$p$- -$p$- [15, 1]. $\mathbb{Z}_{(p)}$ $\mathbb{Z}$ $\mathbb{Z}_{p}$ -$p$- .

2

1) ( -$p$) (2.1) ( -$p$) $G$ , $H_{2}(G,\mathbb{Z}_{(p)})$ .

2) $QR$- $CA$- .

1) (2.1)

[TABLE]

- [12, 2.2] $\mathbb{Z}_{(p)}G$-

[TABLE]

$IG$* - $\mathbb{Z}_{(p)}G$, $(IG)_{G}$ - $G$- $G$- $IG$. $H_{1}(G,IG)\cong H_{2}(G,\mathbb{Z}_{(p)}),$ ,*

[TABLE]

, $R/[R,F]$ $H_{2}(G,\mathbb{Z}_{(p)})$. 1 , - $R/[R,F]$, .

*2) $p$- [14, 3.4] ( $p$- * , $Q_{n},n\geq 1$ [14, 2.4.8]). , $H_{2}(G,\mathbb{Z})$ [14, 2.7.3], $p$- c . [10], $H^{n}(G,\mathbb{Z})$ CA- $n\geq 3$ 2 [10, 2] ( ), $Q_{8}$ 4 [13, 41.1],

[TABLE]

CA- $QR$- .

-$p$- $p$- $G$ ( ), , [3, 2.7], $p$- $G\cong\mathbb{Z}/p^{n}\mathbb{Z}$ ( , ). [15, 1], , , ( ) $p$- $G$ c , ($CA$) .

1

C ( -$p$) $G$ ( -$p$) . , $G$ , .

3

$\mathbb{F}_{p}$- , $\mathbb{F}_{p}$ - $p$ , $(T,t_{0})$ [3, 1.7] $\mathbb{F}_{p}$- $\mathbb{F}_{p}(T,t_{0})$ , $\omega:T\rightarrow\mathbb{F}_{p}(T,t_{0}),\omega(t_{0})=0,$ :

(*) $\gamma:T\rightarrow B$ , $\gamma(t_{0})=0,$ $B$- $\mathbb{F}_{p}$- , $\mathbb{F}_{p}$- $\alpha:\mathbb{F}_{p}(T,t_{0})\rightarrow B,$ $\gamma=\alpha\omega.$

c $\mathbb{F}_{p}$- $\mathbb{F}_{p}(S)$ $S,$ $\mathbb{F}_{p}(S_{\alpha})$, , $T=S\cup\{t_{0}\},$ $t_{0}$ $T$.

$G$ - -$p$- , , $(T,t_{0})$ - $G$ - , $G$ $(T,t_{0})$ , $t_{0}$.

3

[3, 1.8]** $\mathbb{F}_{p}(T,t_{0}),$ $(T,t_{0})$ - $G$- , $G$- , $g\in G$ , () $t\mapsto g\cdot t$ $T$ $T\subset\mathbb{F}_{p}(T,t_{0}).$*

$"$ $"$ - $CA$- . [9, 10, 11] . . [3] ( -$p$- ) -$p$- , $\mathbb{F}_{p}$- .

4

$\mathbb{F}_{p}G$- $\overline{R}/p\overline{R}$ (2.1) $\mathbb{F}_{p}$- , $G$- $\overline{R}/p\overline{R}=R/R^{p}[R,R]\cong\mathbb{F}_{p}(T,t_{0}),$ $(T,t_{0})$- $G$- c .

$\mathbb{F}_{p}$- $\overline{R}/p\overline{R}$ $\mathbb{Z}_{p}$- $\overline{R}$.

5

$\mathbb{Z}_{p}G$- $\overline{R}$ (2.1) $\mathbb{Z}_{p}$- , $G$- $\overline{R}=R/[R,R]\cong\mathbb{Z}_{p}(T,t_{0}),$ $(T,t_{0})$- $G$- c .

$G$ - , $G^{\wedge}_{p}$ -$p$ , , $G^{\wedge}_{p}=\varprojlim_{N\in\mathcal{N}}G/N,$ $\mathcal{N}$ $p$, $\mathcal{N}=\{N\unlhd G,\mid G/N\mid=p^{n},n\in\mathbb{N}\}\cite[cite]{[\@@bibref{}{ZR}{}{}, 2.1.6]}.$ $G$ -$p$- , , $G^{\wedge}_{p}=G.$ ( -$p$) , (2.1), :

[TABLE]

$n\in\mathbb{N},$ $G^{\wedge}_{p}$-

[TABLE]

[TABLE]

$[R,R\mathcal{M}_{n+1}]\subseteq[R,R\mathcal{M}_{n}]$.

$\zeta$- , $p$ $H$ - $p$- $G$. , $H$ $\mathbb{Z}_{p}[\zeta]$ $\xi:H\rightarrow\langle\zeta\rangle$, $\mathbb{Z}_{p}[\zeta]H$ $\mathbb{Z}_{p}[\zeta]$. , $H$ $G$, $\mathbb{Z}_{p}[\zeta]H$ $\mathbb{Z}_{p}[\zeta]G.$

• $G$- *

[TABLE]

$\mathbb{Z}_{p}[\zeta]G$- , $\mathbb{Z}_{p}[\zeta]H$- $\mathbb{Z}_{p}[\zeta]$.

$\pi=1-\zeta,$ $\pi$ $(\pi)$ - $\mathbb{Z}_{p}[\zeta],$ $p\mathbb{Z}_{p}$ ( $(\pi)\cap\mathbb{Z}_{p}=(p)=p\mathbb{Z}_{p}$). $\mathbb{Z}_{p}[\zeta]$ $\mathbb{Z}_{p}$ $p-1,$ $\mathbb{Z}_{p}\hookrightarrow\mathbb{Z}_{p}[\zeta]$ $\mathbb{F}_{p}\cong\mathbb{Z}_{p}[\zeta]/\pi\mathbb{Z}_{p}[\zeta]$ [19, . 7.13].

6

[22, 1]** $G$ - $p$- , , $M$ - $G$- , :

(i) $M$ - $\mathbb{Z}_{p}[\zeta]$- ;

(ii) $M\cong\oplus_{i\in I}\mathbb{Z}_{p}[\zeta]\uparrow^{G}_{H_{i}}$, $M$ .

7

$G$* - -$p$- $\mathfrak{U}$ $G$, $U\lhd G$. , $M=\{M_{U}\}_{U\in\mathfrak{U}}$ - , $\mathbb{Z}_{p}$- $\mathbb{Z}_{p}[G/U]$- $\mathbb{Z}_{p}$- . , $M$ - -$G$- , $U\in\mathfrak{U}$ $\xi_{U}:G/U\rightarrow\langle\zeta\rangle,$ $\zeta$ - $p$- $\mathbb{Z}_{p}$, $G/U$- $M_{U}$ $G/U$- 6.*

-$p$- (2.1) $M_{U}$ , $\overline{R}_{\mathcal{M}_{n}}=R/[R,R\mathcal{M}_{n}]$ $\mathcal{M}_{n}.$ , $M_{U}$ , $\overline{R}_{\mathcal{M}_{n}}=(R/[R,R\mathcal{M}_{n}])^{\wedge}_{p}.$ $p$- $M_{U}$ $p$, , $\overline{R}_{\mathcal{M}_{n}}=(R/[R,R\mathcal{M}_{n}])^{\wedge}_{p}$ $\mathbb{Z}_{p}$- $\mathbb{Z}_{p}[G/\mathcal{M}_{n}]$- $\mathbb{Z}_{p}$- . $\mathcal{T}_{n},\mathcal{T}_{n}^{p}$ , , , , , $\overline{R}=R/[R,R]$ - -$\mathbb{Z}_{p}G^{\wedge}_{p}$- , $\overline{R}/p\overline{R}=R/R^{p}[R,R]$ - -$\mathbb{F}_{p}G^{\wedge}_{p}$- . -$p$ ( - ) -$p$- ( -$p$) , , $p$, - .

, $\mathbb{F}_{p}$- , $R/R^{p}[R,R\mathcal{M}_{n}],$ , $R/[R,R\mathcal{M}_{n}],$ . , $p$- $G$ $M=\mathbb{Z}/p^{k}\mathbb{Z}$ $k\geq 2$ , $p>2$ $\mid Aut(\mathbb{Z}/p^{k}\mathbb{Z})\mid=(p-1)p^{k-1}$. $M$ - ( , , ) , , $\mathbb{Z}_{p}$- , , , $M/pM=\mathbb{Z}/p\mathbb{Z}$ $\mathbb{F}_{p}G$- .

, $\mathcal{T}_{n}$ 7 . , $p>2$ $\mathbb{Z}_{p}$- , $p=2$ $\mathbb{Z}_{2}$ $"$ $"$, 2- , $\pm 1$ $\mp 1$. , $\mathbb{Z}_{p}$ $p>2$ $p$- , $log(1+z):1+p\mathbb{Z}_{p}=U^{(1)}\rightarrow\mathfrak{p}^{1}=p\mathbb{Z}_{p}$ $\mathbb{Z}_{p}$ [19, . 5.5]. $p\mathbb{Z}_{p}$ , $\mathbb{Z}_{p}$. $p=2$ - ( $-1$). , $-1$ $\mathbb{Z}_{2}$. , $\mathbb{Z}_{2}$ . $1+2\mathbb{Z}_{2}\cong(1+4\mathbb{Z}_{2})\times\{\pm 1\}$ ( $1+4\mathbb{Z}_{2}$ $log(1+z):1+4\mathbb{Z}_{2}\rightarrow 4\mathbb{Z}_{2}$), .

[22, 3 ].

1

$M$* $\mathbb{Z}_{p}[\zeta]$- $p$- $G$ $M$. , $\overline{M}=M/\pi M,$ $\pi=1-\zeta,$ $\mathbb{F}_{p}$- $G$- , $M$ $\mathbb{Z}_{p}[\zeta]G$- . $\overline{M}=\oplus_{i\in I}\mathbb{F}_{p}\uparrow^{G}_{H_{i}}$ $M=\oplus_{i\in I}\mathbb{Z}_{p}[\zeta]\uparrow^{G}_{H_{i}}$.*

1

(2.1) - $QR$-( -$p$) , :

) $\mathcal{T}_{n}$ - -$G^{\wedge}_{p}$- ;

) $\mathcal{T}_{n}^{p}$ - -$G^{\wedge}_{p}$- .

)$\Rightarrow$ ) $\mathbb{Z}_{p}[\zeta]$ $\mathbb{Z}_{p}$.

*)$\Rightarrow$ )

• -$p$- , , $p$- $mod(p)$- ( – ). $\overline{R}/p\overline{R}$ - , $\overline{R}/p\overline{R}\cong\oplus_{i\in I}\mathbb{F}_{p}\uparrow^{G}_{H_{i}}.$ , $|I|=dim_{\mathbb{F}_{p}}R/R^{p}[R,F]$, -$p$- [3, 3.2] , $|I|=dim_{\mathbb{F}_{p}}H^{2}(G,\mathbb{F}_{p}),$ $H_{i}$ - $G$. , $n$ (\overline{R}/p\overline{R})_{\mathcal{M}_{n}}=R/R^{p}[R,R\mathcal{M}_{n}]\cong\oplus_{i\in I}\mathbb{F}_{p}\uparrow^{G/\mathcal{M}_{n}}_{H_{i}\mathcal{M}_{n}/\mathcal{M}_{n}}$$R/[R,R\mathcal{M}_{n}]\cong\oplus_{i\in I}\mathbb{Z}_{p}\uparrow^{G/\mathcal{M}_{n}}_{H_{i}\mathcal{M}_{n}/\mathcal{M}_{n}}, , 1. $R/[R,R\mathcal{M}_{n}]\cong\oplus_{i\in I}\mathbb{Z}_{p}\uparrow^{G/\mathcal{M}_{n}}_{H_{i}\mathcal{M}_{n}/\mathcal{M}_{n}},$ ( 6).

, $\mathfrak{p}$- [6, 1.9] $(K_{\mathfrak{p}},\mathcal{O}_{\mathfrak{p}},k),$ $\mathcal{O}_{\mathfrak{p}}$ - 1 ($\mathcal{O}_{\mathfrak{p}}$ - ) $\mathfrak{p}=(\pi)$ $k$ $p>0$, $K_{\mathfrak{p}}$ - 0. -$p$- $\mathcal{T}_{n}=((\frac{R}{[R,R\mathcal{M}_{n}]})^{\wedge}_{p},\phi^{n+1}_{n}),\mathcal{T}_{n}^{p}=(\frac{R}{R^{p}[R,R\mathcal{M}_{n}]},\widetilde{\phi}^{n+1}_{n})$ $p$ $\mathfrak{p}$- $(\mathbb{Q}_{p},\mathbb{Z}_{p},\mathbb{F}_{p})$ . , $"$ $"$ [6, 3.13.3] $\mathbb{F}_{p}\uparrow^{G/\mathcal{M}_{n}}_{H_{i}\mathcal{M}_{n}/\mathcal{M}_{n}}$ ( $\mathbb{F}_{p}$). $\mathbb{Z}_{p}$, , - [6, 1.4], [6, 3.10.2] $R/[R,R\mathcal{M}_{n}]\cong\oplus_{i\in I}\mathbb{Z}_{p}\uparrow^{G/\mathcal{M}_{n}}_{H_{i}\mathcal{M}_{n}/\mathcal{M}_{n}}$ $p$.

2

-$p$- :

1) -$p$- - $p$- ;

2) $\overline{R}=R/[R,R]$ , $\mathbb{F}_{p}$- $mod(p)$ $R/R^{p}[R,R]$.

, -$p$- -$p$- , $CA$- , -$p$- . 1) 2. 1 $\mathbb{Z}_{p}$ $\mathbb{F}_{p}$- , $QR$-( -$p$) 2) 2.

2 $QR$- - . 1) $CA$- .

[3, .71] . . , $\mathbb{Z}_{p}$- -$p$- ( $\mathbb{F}_{p}$- ). 1997 ( -$p$- ). -$p$- ( ) :

1. -$p$- -$p$- ;

2. $\mathbb{Z}_{p}$ $\mathbb{F}_{p}$- .

[15, 16, 17] $p$- . , 2 , . . .

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