Let a and b be two positive integers such that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$a, b < n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. We denote the inclusion <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma \mathbb {C}P^a\rightarrow SU(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mi mathvariant="double-struck">C</mi> <msup> <mi>P</mi> <mi>a</mi> </msup> <mo stretchy="false">→</mo> <mi>S</mi> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon _{a,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. In this article, first, we study the order of the Samelson product <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$\langle \varepsilon _{a,n}, \varepsilon _{b,n}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msub> <mi>ε</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ε</mi> <mrow> <mi>b</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$a+b=n+k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, localized at an odd prime p, we will calculate the order of the commutator map <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="188" /> </InlineMediaObject> <EquationSource Format="TEX">$SU(n)\wedge SU(n) \rightarrow SU(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_521_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$n=4,5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, and continue to give an upper bound on the number of homotopy types of gauge groups for principal SU(4)-and SU(5)-bundles over a n-sphere.

The odd primary Samelson products of SU(n)

期刊论文
发表日期：

2025年11月8日

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Sajjad Mohammadi

摘要

Let a and b be two positive integers such that $a, b < n$ $a, b < n$ . We denote the inclusion $\Sigma \mathbb {C}P^a\rightarrow SU(n)$ $Σ C P^{a} \to S U (n)$ by $\varepsilon _{a,n}$ $ε_{a, n}$ . In this article, first, we study the order of the Samelson product $\langle \varepsilon _{a,n}, \varepsilon _{b,n}\rangle $ $⟨ ε_{a, n}, ε_{b, n} ⟩$ where $a+b=n+k$ $a + b = n + k$ , for $k \ge 0$ $k \geq 0$ . Then, localized at an odd prime p, we will calculate the order of the commutator map $SU(n)\wedge SU(n) \rightarrow SU(n)$ $S U (n) \land S U (n) \to S U (n)$ for $n=4,5$ $n = 4, 5$ , and continue to give an upper bound on the number of homotopy types of gauge groups for principal SU(4)-and SU(5)-bundles over a n-sphere.