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In this chapter we give a geometric representation of \(H_{n}(B;\mathbb {L})\) classes, where \(\mathbb {L}\) is the 4-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes \({\mathcal {N}}^{H}_{n}(B,\partial )\) and the n-th \(\mathbb {L}\) -homology of B, representing the elements of \(H_{n}(B;\mathbb {L})\) by normal degree one maps with a reference map to B. More precisely, we prove that for every \(n \ge 6\) and every finite complex \(B,\) there exists a map \(\Gamma : H_n(B;\mathbb {L}) \longrightarrow \mathcal {N}^{H}_{n}(B,\partial ).\)

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On Geometric Representation of \(\mathbb {L}\) -Homology Classes

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  • 发表日期：

    2025年

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  • Friedrich Hegenbarth,
  • Dušan D. Repovš

摘要

In this chapter we give a geometric representation of \(H_{n}(B;\mathbb {L})\) classes, where \(\mathbb {L}\) is the 4-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes \({\mathcal {N}}^{H}_{n}(B,\partial )\) and the n-th \(\mathbb {L}\) -homology of B, representing the elements of \(H_{n}(B;\mathbb {L})\) by normal degree one maps with a reference map to B. More precisely, we prove that for every \(n \ge 6\) and every finite complex \(B,\) there exists a map \(\Gamma : H_n(B;\mathbb {L}) \longrightarrow \mathcal {N}^{H}_{n}(B,\partial ).\)