Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2022_340_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="192" /> </InlineMediaObject> <EquationSource Format="TEX">$P(2)=P/(\xi_{1}^{p^{2}},\xi_{2}^{p^{2}},\xi_{3}^{p^{2}},\ldots)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mrow> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>1</mn> </mrow> <mrow> <msup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> </mrow> <mrow> <msup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>3</mn> </mrow> <mrow> <msup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2022_340_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">$P=\mathbb{F}_{p}[\xi_{1},\xi_{2},\xi_{3},\ldots]$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>P</mi> <mo>=</mo> <msub> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>ξ</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> </math></EquationSource> </InlineEquation> is the polynomial part of the dual Steenrod algebra and p is an odd prime. In this paper we calculate the cohomology of P(2) in dimensions less than 4 by a May spectral sequence.

The Cohomology of P(2) in Dimensions Less Than 4 at an Odd Prime

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发表日期：

2025年2月25日

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Zhilei Zhang,
Xiangjun Wang,
Linan Zhong

摘要

Let $P(2)=P/(\xi_{1}^{p^{2}},\xi_{2}^{p^{2}},\xi_{3}^{p^{2}},\ldots)$ $P (2) = P / (ξ_{1}^{p^{2}}, ξ_{2}^{p^{2}}, ξ_{3}^{p^{2}}, \dots)$ , where $P=\mathbb{F}_{p}[\xi_{1},\xi_{2},\xi_{3},\ldots]$ $P = F_{p} [ξ_{1}, ξ_{2}, ξ_{3}, \dots]$ is the polynomial part of the dual Steenrod algebra and p is an odd prime. In this paper we calculate the cohomology of P(2) in dimensions less than 4 by a May spectral sequence.