<p>In this paper, we examine the question raised by Valerie Bardakov concerning the lifting and extension problem of the exact sequence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="210" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \rightarrow U_{n+1} \rightarrow P_{n+1} \rightarrow P_n \rightarrow 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>U</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 4.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We show that, for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists an automorphism of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> which cannot be lifted to an automorphism of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. In addition, we show that there exist an automorphism of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$U_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> which cannot be extended to an automorphism of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1207_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>.</p>

On the lifting and extension problems of the automorphisms of the Pure braid group

期刊论文
发表日期：

2025年3月19日

查看全文

Rana S. Kahil,
Mohammad N. Abdulrahim

摘要

In this paper, we examine the question raised by Valerie Bardakov concerning the lifting and extension problem of the exact sequence $1 \rightarrow U_{n+1} \rightarrow P_{n+1} \rightarrow P_n \rightarrow 1$ $1 \to U_{n + 1} \to P_{n + 1} \to P_{n} \to 1$ for $n\ge 4.$ $n \geq 4 .$ We show that, for $n\ge 4$ $n \geq 4$ , there exists an automorphism of $P_{n}$ $P_{n}$ which cannot be lifted to an automorphism of $P_{n+1}$ $P_{n + 1}$ . In addition, we show that there exist an automorphism of $U_{n+1}$ $U_{n + 1}$ which cannot be extended to an automorphism of $P_{n+1}$ $P_{n + 1}$ .