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<p>The cohomological classification of orbit spaces of free <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(G=\mathbb {Z}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(G=\mathbb {S}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> actions on the mod 2 or rational cohomology product of three spheres <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {S}^n\times \mathbb {S}^m \times \mathbb {S}^l,1\le n\le m\le l \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>m</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>l</mi> </msup> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>l</mi> </mrow> </math></EquationSource> </InlineEquation>, has been discussed in [<CitationRef CitationID="CR5">5</CitationRef>, <CitationRef CitationID="CR14">14</CitationRef>]. In this paper, we discuss orbit spaces of free <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(G=\mathbb {S}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> actions on a finitistic space <i>X</i> whose rational cohomology is isomorphic to the product of three spheres. As an application, we determine Borsuk-Ulam type theorems.</p>

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Classification of orbit spaces of free \(\mathbb {S}^3\)- actions on the product of three spheres

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • Dimpi,
  • Jitendra Kumar Maitra

摘要

The cohomological classification of orbit spaces of free \(G=\mathbb {Z}_2\) G = Z 2 or \(G=\mathbb {S}^1\) G = S 1 actions on the mod 2 or rational cohomology product of three spheres \(\mathbb {S}^n\times \mathbb {S}^m \times \mathbb {S}^l,1\le n\le m\le l \) S n × S m × S l , 1 ≤ n ≤ m ≤ l , has been discussed in [5, 14]. In this paper, we discuss orbit spaces of free \(G=\mathbb {S}^3\) G = S 3 actions on a finitistic space X whose rational cohomology is isomorphic to the product of three spheres. As an application, we determine Borsuk-Ulam type theorems.