关于连续映射同伦纤维hfib(f,y₀)的释义及符号说明问询

Hey there! Let's break down exactly what hfib(f,y₀) means when we're working with continuous maps in algebraic topology.

First, a quick recap of the setup: we've got a continuous map f: X → Y, and a fixed point y₀ ∈ Y. The notation hfib(f,y₀) stands for the homotopy fiber of f over the point y₀. Here's what that actually entails:

Core Definition #

The homotopy fiber is a topological space constructed to capture the "homotopy-theoretic fiber" of the map f, even when f isn't a fiber bundle (where we'd have a standard, literal fiber). Formally, we define:
hfib(f,y₀) = {(x, γ) | x ∈ X, γ: [0,1] → Y is continuous, γ(0) = f(x), γ(1) = y₀}
We equip this set with the compact-open topology (a standard topology for function spaces that ensures continuity plays nicely with our construction).

Intuition & Key Properties #

• Think of it as all points x ∈ X paired with a continuous path that "pulls" f(x) back to y₀ in Y. It's like a generalized fiber that accounts for paths/homotopies, not just preimages.
• If f is a fiber bundle, then the ordinary fiber f⁻¹(y₀) (all x where f(x)=y₀) is homotopy equivalent to hfib(f,y₀)—this is where the "homotopy" in the name comes from; it preserves the essential homotopy type of the fiber even when f isn't a bundle.
• The homotopy type of hfib(f,y₀) only depends on the homotopy class of y₀ in Y. That means if y₀ and y₁ can be connected by a continuous path in Y, their homotopy fibers over f will be homotopy equivalent.

备注：内容来源于stack exchange，提问作者Ziqiang Cui