<p>We study smooth, closed orientable <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$S^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-manifolds <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> with exactly <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> fixed points. We show that the dimension of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> is of the form <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$4\cdot 2^a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>·</mo> <msup> <mn>2</mn> <mi>a</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$8\cdot (2^a+2^b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>8</mn> <mo>·</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>a</mi> </msup> <mo>+</mo> <msup> <mn>2</mn> <mi>b</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$a,b\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$a\ne b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, under the extra assumption that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1230_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> is spin or unitary, we determine exactly the possible dimensions of M.</p>

On circle actions with exactly three fixed points

期刊论文
出版模式：

开放获取
发表日期：

2025年8月11日

查看全文

Michael Wiemeler

摘要

We study smooth, closed orientable $S^1$ $S^{1}$ -manifolds $M$ $M$ with exactly $3$ $3$ fixed points. We show that the dimension of $M$ $M$ is of the form $4\cdot 2^a$ $4 \cdot 2^{a}$ or $8\cdot (2^a+2^b)$ $8 \cdot (2^{a} + 2^{b})$ with $a,b\ge 0$ $a, b \geq 0$ and $a\ne b$ $a \neq b$ . Moreover, under the extra assumption that $M$ $M$ is spin or unitary, we determine exactly the possible dimensions of M.