本科高年级Geometry & Topology计划中

Homotopy and the fundamental group

Homotopy of maps and paths, the fundamental group, covering spaces, the computation π₁(S¹) ≅ ℤ, the Seifert–van Kampen theorem and applications such as the two-dimensional Brouwer fixed-point theorem.

math.geom.fundamental-group

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前置知识

Metric spaces and point-set topology Groups

40 项必需的前置知识（一直追溯到底）

你将学会

Define homotopy and the fundamental group, and prove that the fundamental group is a homotopy invariant.
Compute π₁(S¹) ≅ ℤ using covering spaces and deduce the Brouwer fixed-point theorem for the disc.
Use the Seifert–van Kampen theorem to compute fundamental groups of wedges of circles and of surfaces.

定义

A homotopy between maps f, g : X → Y is a continuous map H : X × [0, 1] → Y with H(x, 0) = f(x) and H(x, 1) = g(x).
π₁(X, x₀) is the group of homotopy classes of loops based at x₀, with concatenation as the operation.
A covering map p : E → B is a continuous surjection such that each point of B has an open neighbourhood U whose preimage is a disjoint union of open sets, each mapped homeomorphically onto U.

定理

π₁(S¹) ≅ ℤ.
Brouwer: every continuous map from the closed disc to itself has a fixed point.
The fundamental group of a wedge of two circles is the free group on two generators.
For path-connected, locally path-connected and semilocally simply connected spaces, connected covering spaces correspond to conjugacy classes of subgroups of π₁.

常见误解

Thinking fundamental groups are always abelian.
Believing spaces with isomorphic fundamental groups are homeomorphic; S² and a point both have trivial fundamental group.

通向

Algebraic topology

可选发现

Homotopy type theory In homotopy type theory the circle is a higher inductive type, and π₁(S¹) ≅ ℤ has a synthetic proof.
← Functors

学习路径

Graduate preparation