Difference between revisions of "Category:Algebraic Topology"

A

• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell
• A compact and convex subset of Euclidean n-space with non-empty interior is a closed n-cell and its interior is an open n-cell/Statement
• A continuous map induces a homomorphism between fundamental groups
• A continuous map induces a homomorphism on fundamental groups
• Abstract simplicial complex

C

• Closed n-cell
• Concatenation of loops and paths (homotopy)
• Contractible topological space
• Covering map (topology)
• Covering map (topology)/Definition
• Covering space
• Covering space (topology)
• CW-complex
• Notes:CW-Complex

D

• Deformation retract
• Deformation retraction
• Deformation retraction/Definition

E

• End point preserving homotopic
• End-point-preseriving homotopic
• End-point-preserving homotopic paths
• Even covering (topology)
• Evenly covered (topology)
• Evenly covered by a continuous map
• Evenly covers (topology)

F

• First homotopy group
• Fundamental group homomorphism induced by a continuous map
• Fundamental groups

G

• Geometric independence
• Geometrically independent
• Geometrically independent set

H

• Homeomorphic topological spaces have isomorphic fundamental groups
• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
• Homotopic maps
• Homotopy
• Homotopy concatenation
• Homotopy equivalence
• Homotopy equivalent topological space
• Homotopy equivalent topological spaces
• Homotopy invariance of loop concatenation
• Homotopy invariance of path concatenation
• Homotopy of maps is an equivalence relation

I

• Induced homomorphism on fundamental groups

L

• Lifting of a continuous map through a covering map
• Locally path connected (topology)
• Locally path connected topological space
• Locally path-connected (topology)
• Locally path-connected topological space

N

• N-cell
• N-plane
• Nth homotopy group

O

• Omega(X,b)
• Open n-cell

P

• Path concatenation (topology)
• Path connected (topology)
• Path connected topological space
• Path-connected (topology)
• Path-connected topological space
• Proof that the fundamental group is actually a group
• Proof that the fundamental group is actually a group/Outline

R

• Retract (topology)

S

• Simplex
• Simplicial complex
• Simply connected topological space
• Square lemma (of homotopic paths)
• Stages of a homotopy
• Strong deformation retract

T

• The composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths
• The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms
• The induced fundamental group homomorphism of a composition of continuous maps is the same as the composition of their induced homomorphisms/Statement
• The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group
• The induced fundamental group homomorphism of the identity map is the identity map of the fundamental group/Statement
• The relation of path-homotopy is preserved under composition with continuous maps
• The singular homology groups of a 1-point space are all trivial except the zeroth which is isomorphic to the integers
• The stages of a homotopy are continuous
• Topological covering space
• Topological retraction
• Topological retraction/Definition
• Torus
• Trivial loop
• Trivial loop based at
• Trivial loop based at a point
• Notes:Types of retractions
• Types of topological retractions

U

• Unique lifting property

V

• Vertex scheme (abstract simplicial complex)
• Vertex scheme of an abstract simplicial complex
• Vertex scheme of an abstract simplicial complex/Definition
• Vertex set (abstract simplicial complex)
• Vertex set of an abstract simplicial complex
• Vertex set of an abstract simplicial complex/Definition