The homomorphism on fundamental groups induced by a pointed continuous map

Depends on

• Based loops and the fundamental group
• Loop classes form the group $\pi_1(X,x_0)$ under concatenation
• Continuity of a map of topological spaces at a point and globally
• Monoid homomorphism and group homomorphism

Used by

• Finiteness of the fundamental group is sufficient, but not necessary, for Reeb stability Corollary
• Pure geometric braids and ordered configuration loops Corollary
• Reebless leaves are pi-one-injective and transverse loops are essential Corollary
• A noncompact foliation violating Novikov's compactness conclusions Counterexample
• Equal abelianizations do not classify knots Counterexample
• Semilocally simply connected spaces with explicit basepoint convention Definition
• The configuration braid group Bₙᶜᵒⁿᶠ as the fundamental group of an unordered configuration space Definition
• The holonomy representation and the holonomy group of a leaf Definition
• The one-variable Alexander module of an oriented link Definition
• The pure braid group PBₙ as the fundamental group of an ordered configuration space Definition
• Bridge and unknotting numbers of the trefoil Example
• One-surgery on a three-manifold as framed knot surgery Example
• A crossing-ball cover of a knot complement Lemma
• Braids lift to the LKB cover and act Lambda-linearly Lemma
• Changing the point over a fixed basepoint conjugates the induced covering subgroup Lemma
• Homotopic-loop factorizations have the same value in the group pushout Lemma
• In a transversely oriented codimension-one foliation a compact leaf with finite fundamental group has trivial holonomy Lemma
• Loops over a two-set path-connected open cover factor through the covering sets Lemma
• Removing the point at infinity preserves the knot group Lemma
• The combed geometric decomposition is unique Lemma
• The deck group of the holonomy cover is the holonomy group Lemma
• The fundamental-group label controls contractibility of the Whitney circle Lemma
• The interior-disc and closed-disc configuration spaces are homotopy equivalent Lemma
• A based morphism between connected coverings exists exactly when the induced subgroups are included Proposition
• The fundamental group is a functor π₁:Top_*toGrp Proposition
• The geometric endpoint permutation matches covering monodromy Proposition
• A covering map induces an injective homomorphism on fundamental groups Theorem
• Correctness of the Wirtinger presentation Theorem
• Geometric braid classes and the unordered configuration fundamental group Theorem
• Induced fundamental-group maps are well defined, functorial and invariant under based homotopy Theorem
• Lifting criterion for maps from path-connected locally path-connected spaces Theorem
• Novikov's Reeb component theorem Theorem
• Seifert–van Kampen identifies the fundamental group with a group pushout Theorem
• The configuration braid short exact sequence 1→ PBₙ→ Bₙᶜᵒⁿᶠ→ Sₙ→ 1 Theorem
• The conformal parameter of a round annulus is a complete invariant Theorem
• The Davis complex is simply connected Theorem
• π₁(X× Y,(x₀,y₀))≅π₁(X,x₀)×π₁(Y,y₀) Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• A. Hatcher, Algebraic Topology, Chapter 1, Induced Homomorphisms (standard reference, not scraped)