query: Possible rank of closed orientable manifolds
positive_passages.docid: 155390
positive_passages.text: Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?
negative_passages.docid: 56646
negative_passages.text: Arithmetic progression topology
negative_passages.docid: 122352
negative_passages.text: Topologies in a Riemannian Manifold
negative_passages.docid: 71791
negative_passages.text: Smooth structure of embedded submanifolds
negative_passages.docid: 156245
negative_passages.text: What does it mean for the group of rotation matrices to have a "manifold structure"
negative_passages.docid: 61446
negative_passages.text: Fixed point of interior of closed disk
negative_passages.docid: 55064
negative_passages.text: Homology groups of torus
negative_passages.docid: 36519
negative_passages.text: Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature
negative_passages.docid: 147793
negative_passages.text: A variation on the induced topology (subspace topology)
negative_passages.docid: 90365
negative_passages.text: Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group
negative_passages.docid: 154142
negative_passages.text: Finding a specific basis for an endomorphism
negative_passages.docid: 151581
negative_passages.text: Prerequisite for Milnor Topology from a Differential Viewpoint
negative_passages.docid: 136108
negative_passages.text: Arbitrary intersection of closed, connected subsets of a compact space connected?
negative_passages.docid: 98225
negative_passages.text: Dimensions with Rank-Nullity question
negative_passages.docid: 145339
negative_passages.text: Index of the subgroup $\operatorname{Im}(f)$
negative_passages.docid: 123430
negative_passages.text: Can a topological manifold be non-connected and each component with different dimension?
negative_passages.docid: 100437
negative_passages.text: Classification of connected subsets of the real line (up to homeomorphism)