Splitting the Madsen-Tillmann Spectra $MTθ_n$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929758502125568 |
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| author | Pedersen, Jonathan Sejr Senger, Andrew |
| author_facet | Pedersen, Jonathan Sejr Senger, Andrew |
| contents | We prove that the Madsen-Tillmann spectrum $MTθ_n$ splits into the sum of spectra $Σ^{-2n}MO\langle n+1 \rangle \oplus Σ^{\infty-2n}\mathbb{R} P^\infty_{2n}$ after Postnikov trunctation $τ_{\leq \ell}$ for $\ell = \lfloor \frac{n}{2} \rfloor - 6$. To accomplish this, we prove that the connecting map in a certain fiber sequence is nullhomotopic in this range by an Adams filtration argument. As an application, we compute $H_2(B\operatorname{Diff}(W^{2n}_{g},D^{2n});\mathbb{Z})$ up to extensions for $n \geq 16$ and $g \geq 7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting the Madsen-Tillmann Spectra $MTθ_n$ Pedersen, Jonathan Sejr Senger, Andrew Algebraic Topology We prove that the Madsen-Tillmann spectrum $MTθ_n$ splits into the sum of spectra $Σ^{-2n}MO\langle n+1 \rangle \oplus Σ^{\infty-2n}\mathbb{R} P^\infty_{2n}$ after Postnikov trunctation $τ_{\leq \ell}$ for $\ell = \lfloor \frac{n}{2} \rfloor - 6$. To accomplish this, we prove that the connecting map in a certain fiber sequence is nullhomotopic in this range by an Adams filtration argument. As an application, we compute $H_2(B\operatorname{Diff}(W^{2n}_{g},D^{2n});\mathbb{Z})$ up to extensions for $n \geq 16$ and $g \geq 7$. |
| title | Splitting the Madsen-Tillmann Spectra $MTθ_n$ |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2503.10507 |