On geometric representation of $\mathbb{L}$-homology classes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913959512113152 |
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| author | Hegenbarth, Friedrich Repovš, Dušan D. |
| author_facet | Hegenbarth, Friedrich Repovš, Dušan D. |
| contents | 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 $Γ: H_n(B;\mathbb{L}) \longrightarrow \mathcal{N}^{H}_{n}(B,\partial).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On geometric representation of $\mathbb{L}$-homology classes Hegenbarth, Friedrich Repovš, Dušan D. Algebraic Topology Geometric Topology Primary: 55R20, 57P10, 57R65, 57R67, Secondary: 55M05, 55N99, 57P05, 57P99 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 $Γ: H_n(B;\mathbb{L}) \longrightarrow \mathcal{N}^{H}_{n}(B,\partial).$ |
| title | On geometric representation of $\mathbb{L}$-homology classes |
| topic | Algebraic Topology Geometric Topology Primary: 55R20, 57P10, 57R65, 57R67, Secondary: 55M05, 55N99, 57P05, 57P99 |
| url | https://arxiv.org/abs/2503.17972 |