Algebraic Topology

Year 4 · 4 sections · 20 concepts · 67 practice questions

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What's covered

free1 · Homotopy4 concepts

Definition 1.1 · Homotopy of maps
Proposition 1.2 · Straight-line homotopies and maps into convex sets
Definition 1.3 · Homotopy equivalence and contractible spaces
Definition 1.4 · Retracts and deformation retracts

2 · The fundamental group6 concepts

Definition 2.1 · Path homotopy and concatenation
Theorem 2.2 · The fundamental group
Proposition 2.3 · Change of basepoint
Theorem 2.4 · Induced homomorphisms and homotopy invariance
Proposition 2.5 · Simply connected spaces; spheres of dimension at least 2
Proposition 2.6 · Fundamental group of a product

3 · Covering spaces and π₁(S¹)5 concepts

Definition 3.1 · Covering maps
Lemma 3.2 · Path lifting
Lemma 3.3 · Homotopy lifting
Theorem 3.4 · π₁(S¹) ≅ ℤ
Definition 3.5 · Degree of a map S¹ → S¹

4 · Applications of π₁(S¹)5 concepts

Theorem 4.1 · No retraction of the disc onto the circle
Theorem 4.2 · Brouwer's fixed point theorem (dimension 2)
Theorem 4.3 · The fundamental theorem of algebra
Theorem 4.4 · The Borsuk–Ulam theorem (dimension 2)
Corollary 4.5 · ℝ² is not homeomorphic to ℝⁿ for n ≠ 2

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Definition 1.1 · Homotopy of maps

Homotopy: a continuous deformation $f_t$ from $f = f_0$ to $g = f_1$ , i.e. one map $X\times I\to Y$ . Rel $A$ : points of $A$ stay put throughout. Key tool: gluing homotopies end to end uses the pasting lemma. Tiny example: any two maps $X\to\mathbb{R}^n$ are homotopic via $(1 - t)f + tg$ .

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