Category:Homotopy Theory

A

• A continuous map induces a homomorphism between fundamental groups
• A continuous map induces a homomorphism on fundamental groups

C

• Concatenation of loops and paths (homotopy)
• Concatenation of paths and loops (homotopy)
• Constant loop
• Constant loop based at
• Constant loop based at a point
• Contractible topological space
• Covering map (topology)
• Covering map (topology)/Definition
• Covering space
• Covering space (topology)
• 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

F cont.

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

H

• Homeomorphic topological spaces have isomorphic fundamental groups
• Homeomorphic topological spaces have isomorphic fundamental groups/Statement
• Homotopic
• Homotopic maps
• Homotopy concatenation
• Homotopy equivalence
• Homotopy equivalent topological space
• Homotopy equivalent topological spaces
• Homotopy invariance of loop concatenation
• Homotopy invariance of path concatenation
• Homotopy is an equivalence relation on the set of all continuous maps between spaces
• Homotopy of maps is an equivalence relation
• HomotopyPage

I

• Induced homomorphism on fundamental groups

L

• Lifting of a continuous map through a covering map
• Loop
• Loop concatenation
• Loops

O

• Omega(X,b)

P

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

R

• Retract (topology)

S

• 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 fundamental group
• 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 maps being homotopic is an equivalence relation
• The relation of path-homotopy is preserved under composition with continuous maps
• Topological covering space
• Topological retraction
• Topological retraction/Definition
• Torus
• Trivial loop
• Trivial loop based at
• Trivial loop based at a point
• Types of topological retractions

U

• Unique lifting property