nLab
homotopy relative boundary

For paths

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

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For paths

Definition

Let XX be a topological space and let

γ 1,γ 2:[0,1]X \gamma_1, \gamma_2 \;\colon\; [0,1] \longrightarrow X

be two paths in XX, i.e. two continuous functions from the closed interval to XX, such that their endpoints agree:

γ 1(0)=γ 2(0)AAAAγ 1(1)=γ 2(1). \gamma_1(0) = \gamma_2(0) \phantom{AAAA} \gamma_1(1) = \gamma_2(1) \,.

Then a homotopy relative boundary from γ 1\gamma_1 to γ 2\gamma_2 is a homotopy (def. )

η:γ 1γ 2 \eta \;\colon\; \gamma_1 \Rightarrow \gamma_2

such that it does not move the endpoints:

η(0,)=const γ 1(0)=const γ 2(0)AAAAAAη(1,)=const γ 1(1)=const γ 2(1). \eta(0,-) = const_{\gamma_1(0)} = const_{\gamma_2(0)} \phantom{AAAAAA} \eta(1,-) = const_{\gamma_1(1)} = const_{\gamma_2(1)} \,.
Proposition

Let XX be a topological space and let x,yXx, y \in X be two points. Write

P x,yX P_{x,y} X

for the set of paths γ\gamma in XX with γ(0)=x\gamma(0) = x and γ(1)=y\gamma(1) = y.

Then homotopy relative boundary (def. ) is an equivalence relation on P x,yXP_{x,y}X.

The corresponding set of equivalence classes is denoted

Hom Π 1(X)(x,y)(P x,yX)/. Hom_{\Pi_1(X)}(x,y) \;\coloneqq\; (P_{x,y}X)/\sim \,.

The operation of path concatenation descends to these equivalence classes, so that for all x,y,zXx,y, z \in X there is a function

Hom Π 1(X)(x,y)×Hom Π 1(X)(y,z)Hom Π 1(X)(x,z). Hom_{\Pi_1(X)}(x,y) \times Hom_{\Pi_1(X)}(y,z) \longrightarrow Hom_{\Pi_1(X)}(x,z) \,.

Moreover, this composition operation is associative in the evident sense.

Set of points of XX together with the set Hom Π 1(X)(x,y)Hom_{\Pi_1(X)}(x,y) for all points of points and equipped with this composition operation is called the fundamental groupoid of XX, denoted

Π 1(X). \Pi_1(X) \,.

If we pick a single point xXx \in X, then one writes

π 1(X,x)Hom Π 1(X)(x,x). \pi_1(X,x) \;\coloneqq\; Hom_{\Pi_1(X)}(x,x) \,.

Under the above composition this is a group, and as such is called the fundamental group of XX at xx.

Last revised on September 17, 2019 at 06:41:12. See the history of this page for a list of all contributions to it.