Connectivity of finite subset spaces of cell complexes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2003
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| _version_ | 1866916331690917888 |
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| author | Tuffley, Christopher |
| author_facet | Tuffley, Christopher |
| contents | The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth finite subset space of a connected cell complex is (k-2)-connected, and (k-1)-connected if in addition the underlying space is simply connected. We expect exp_k X to be (k+m-2)-connected if X is an m-connected cell complex, and reduce proving this to the problem of proving it for finite wedges of (m+1)-spheres. Our results complement a theorem due to Handel that for path-connected Hausdorff X the map on pi_i induced by the inclusion exp_k X --> exp_{2k+1} X is zero for all k and i. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0304086 |
| institution | arXiv |
| publishDate | 2003 |
| record_format | arxiv |
| spellingShingle | Connectivity of finite subset spaces of cell complexes Tuffley, Christopher Geometric Topology 55R80 (54B20 55Q52) The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth finite subset space of a connected cell complex is (k-2)-connected, and (k-1)-connected if in addition the underlying space is simply connected. We expect exp_k X to be (k+m-2)-connected if X is an m-connected cell complex, and reduce proving this to the problem of proving it for finite wedges of (m+1)-spheres. Our results complement a theorem due to Handel that for path-connected Hausdorff X the map on pi_i induced by the inclusion exp_k X --> exp_{2k+1} X is zero for all k and i. |
| title | Connectivity of finite subset spaces of cell complexes |
| topic | Geometric Topology 55R80 (54B20 55Q52) |
| url | https://arxiv.org/abs/math/0304086 |