The Gerstenhaber algebra structure on the rational loop homology <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mathbb {H}_*(LX,\mathbb {Q}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi mathvariant="double-struck">H</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mi>X</mi> <mo>,</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a simply connected compact manifold X can be determined in terms of derivations of the minimal Sullivan model of X. Using this result we compute the Gerstenhaber algebra structure on the rational loop homology of quaternionic Stiefel manifolds <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ V_{n,k}(\mathbb {H}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Further, we compute the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ S^1 $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-equivariant cohomology of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ LV_{n,k}(\mathbb {H}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <msub> <mi>V</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

String topology of quaternionic Stiefel manifolds

期刊论文
发表日期：

2025年8月19日

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Rampao K. Sangma,
Angom Tiken Singh

摘要

The Gerstenhaber algebra structure on the rational loop homology $ \mathbb {H}_*(LX,\mathbb {Q}) $ $H_{*} (L X, Q)$ of a simply connected compact manifold X can be determined in terms of derivations of the minimal Sullivan model of X. Using this result we compute the Gerstenhaber algebra structure on the rational loop homology of quaternionic Stiefel manifolds $ V_{n,k}(\mathbb {H}) $ $V_{n, k} (H)$ . Further, we compute the $ S^1 $ $S^{1}$ -equivariant cohomology of $ LV_{n,k}(\mathbb {H}) $ $L V_{n, k} (H)$ .