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Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

Mësime

Advanced The fundamental group Loops up to deformation; the circle's winding number; van Kampen. polar form of -1 Advanced Homology and the Euler characteristic Counting holes in every dimension with chain complexes. 6 - 12 + 8

Attach a group to a space so that continuous maps become homomorphisms. Then the question "are these two spaces the same shape?" becomes an algebra computation, which is how we know a doughnut is not a sphere.

Mësime

Homotopy

Algebraic topologyHomotopyCovering spaceSeifert–Van Kampen theoremHomotopy groupSimply connected spaceDeformation retract

Homology

Homology (mathematics)Simplicial complexSingular homologyChain complexExact sequenceBetti numberEuler characteristicCW complexMayer–Vietoris sequenceCohomologyPoincaré duality

Consequences

Brouwer fixed-point theoremBorsuk–Ulam theoremHairy ball theoremJordan curve theoremPoincaré conjectureClassification of manifolds

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What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

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