Intertwining category and complexity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912838930399232 |
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| author | Jauhari, Ekansh |
| author_facet | Jauhari, Ekansh |
| contents | We develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space $X$, denoted by $\mathsf{icat}(X)$ and $\mathsf{iTC}_m(X)$, respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts $\mathsf{dcat}(X)$ and $\mathsf{dTC}_m(X)$, and their classical counterparts $\mathsf{cat}(X)$ and $\mathsf{TC}_m(X)$, such as homotopy invariance and special behavior on topological groups. We show that the notions of $\mathsf{iTC}_m$ and $\mathsf{dTC}_m$ are different for each $m \ge 2$ by proving that $\mathsf{iTC}_m(\mathcal{H})=1$ for all $m \ge 2$ for Higman's group $\mathcal{H}$. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes $X$ for which $\mathsf{icat}(X) > 1$, $\mathsf{iTC}_m(X) > 1$, $\mathsf{icat}(X) = \mathsf{dcat}(X) = \mathsf{cat}(X)$, and $\mathsf{iTC}(X) = \mathsf{dTC}(X) = \mathsf{TC}(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12265 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intertwining category and complexity Jauhari, Ekansh Algebraic Topology 55M30 (Primary) 60B05, 54A10, 68T40 (Secondary) We develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space $X$, denoted by $\mathsf{icat}(X)$ and $\mathsf{iTC}_m(X)$, respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts $\mathsf{dcat}(X)$ and $\mathsf{dTC}_m(X)$, and their classical counterparts $\mathsf{cat}(X)$ and $\mathsf{TC}_m(X)$, such as homotopy invariance and special behavior on topological groups. We show that the notions of $\mathsf{iTC}_m$ and $\mathsf{dTC}_m$ are different for each $m \ge 2$ by proving that $\mathsf{iTC}_m(\mathcal{H})=1$ for all $m \ge 2$ for Higman's group $\mathcal{H}$. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes $X$ for which $\mathsf{icat}(X) > 1$, $\mathsf{iTC}_m(X) > 1$, $\mathsf{icat}(X) = \mathsf{dcat}(X) = \mathsf{cat}(X)$, and $\mathsf{iTC}(X) = \mathsf{dTC}(X) = \mathsf{TC}(X)$. |
| title | Intertwining category and complexity |
| topic | Algebraic Topology 55M30 (Primary) 60B05, 54A10, 68T40 (Secondary) |
| url | https://arxiv.org/abs/2406.12265 |