Book 7A

Topology and the Fundamental Group

What the Poincaré conjecture is about

Topological spaces and quotients, manifolds and surfaces, the fundamental group, covering spaces, smooth topology, a little homology, and the precise statement of the Poincaré conjecture.

Main companionIntroduction to Topological Manifolds · John M. Lee
Main companionAlgebraic Topology · Allen Hatcher
Main companionTopology from the Differentiable Viewpoint · John Milnor

In preparation

1
Topological Spaces and Quotients
What changes beyond metric spaces, and gluing spaces together.

In preparation
2
Compactness and Compactification
Hausdorff spaces, Tychonoff, and the sphere as ℝⁿ plus a point.

In preparation
3
Manifolds and Surfaces
Topological manifolds, the classification of surfaces and the Euler characteristic.

In preparation
4
The Fundamental Group
Loops up to deformation, and π₁ of the circle.

In preparation
5
Computing π₁
Seifert–van Kampen, surfaces, projective spaces and knots.

In preparation
6
Covering Spaces
Lifting, universal covers, lens spaces and the Poincaré homology sphere.

In preparation
7
Smooth Topology
Sard’s theorem, degree, the hairy ball theorem and Poincaré–Hopf.

In preparation
8
Homology in Brief
Why homology is not enough, and why π₁ is the right invariant.

In preparation
9
The Poincaré Conjecture, Precisely
Every word of the statement, and why each hypothesis is needed.

In preparation
10
Morse Theory
Optional: critical points, handles and Heegaard splittings.

In preparation

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Topology and the Fundamental Group

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