A structured visual guide to the major mathematical areas and their relationships.
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This subtopic introduces the core ideas in topological manifolds, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Topology of $E^2$, $2$-manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of general $3$-manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of $E^3$ and $S^3$. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of $E^4$, $4$-manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of $E^n$, $n$-manifolds ($4 < n < \infty$). This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Geometric structures on manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of topological fiber spaces. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Topology of infinite-dimensional manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Shapes. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Engulfing in topological manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Embeddings and immersions of manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Isotopy and pseudo-isotopy. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Connected sums. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Flatness and tameness of topological manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
$S^1$-actions and related topics. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Microbundles and block bundles (MSC2000). This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Cobordism and concordance in topological manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Algebraic topology of manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Cobordism and concordance in topological manifolds (MSC2000). This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
General position and transversality in topological manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.
Stratifications of topological manifolds. This topic develops the theory of topological manifolds, including local and global structure, decomposition, smoothing relations, and classification issues.
Wikipedia: Topological manifold
Used to formalize spaces that are locally Euclidean while allowing broad global topological complexity.