Notes On The Course Algebraic Topology by Boris Botvinnik

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Notes On The Course Algebraic Topology written by Boris Botvinnik This is an other great mathematics book cover the following topics.

Important examples of topological spaces
Euclidian space, spheres, disks, Real projective spaces, Complex projective spaces, Grassmannian manifolds, Flag manifolds, Classic Lie groups, Stiefel manifolds, Surfaces

Constructions
Product, Cylinder, suspension, Glueing, Join, Spaces of maps, loop spaces, path spaces, Pointed spaces

Homotopy and homotopy equivalence
Definition of a homotopy, Homotopy classes of maps, Homotopy equivalence., Retracts, The case of “pointed” spaces

CW -complexes
Basic definitions, Some comments on the definition of a CW -complex, Operations on CW -complexes, More examples of CW -complexes, CW -structure of the Grassmanian manifolds

CW -complexes and homotopy
Borsuk’s Theorem on extension of homotopy, Cellular Approximation Theorem, Completion of the proof of Theorem, Fighting a phantom: Proof of Lemma, Back to the Proof of Lemma, First applications of Cellular Approximation Theorem

Fundamental group
General definitions, One more definition of the fundamental group, Dependence of the fundamental group on the base point, Fundamental group of circle, Fundamental group of a finite CW -complex, Theorem of Seifert and Van Kampen

Covering spaces
Definition and examples, Theorem on covering homotopy, Covering spaces and fundamental group, Observation, Lifting to a covering space, Classification of coverings over given space, Homotopy groups and covering spaces, Lens spaces

Higher homotopy groups
More about homotopy groups, Dependence on the base point, Relative homotopy groups

Fiber bundles
First steps toward fiber bundles, Constructions of new fiber bundles, Serre fiber bundles, Homotopy exact sequence of a fiber bundle, More on the groups pn(X, A; x0)

Suspension Theorem and Whitehead product
The Freudenthal Theorem, First applications, A degree of a map Sn ?, Stable homotopy groups of spheres, Whitehead product

Homotopy groups of CW -complexes
Changing homotopy groups by attaching a cell, Homotopy groups of a wedge, The first nontrivial homotopy group of a CW -complex, Weak homotopy equivalence, Cellular approximation of topological spaces, Eilenberg-McLane spaces, Killing the homotopy groups

Homology groups: basic constructions
Singular homology, Chain complexes, chain maps and chain homotopy, First computations, Relative homology groups, Relative homology groups and regular homology groups, Excision Theorem, Mayer-Vietoris Theorem

Homology groups of CW -complexes
Homology groups of spheres, Homology groups of a wedge, Maps g, Cellular chain complex, Geometric meaning of the boundary homomorphism ?, Some computations, Homology groups of RPn, Homology groups of CPn, HPn

Homology and homotopy groups
Homology groups and weak homotopy equivalence, Hurewicz homomorphism, Hurewicz homomorphism in the case n = 1, Relative version of the Hurewicz Theorem

Homology with coefficients and cohomology groups
Definitions, Basic propertries of H*(-; G) and H*(-; G), Coefficient sequences, The universal coefficient Theorem for homology groups, The universal coefficient Theorem for cohomology groups, The K¨unneth formula, The Eilenberg-Steenrod Axioms

Some applications
The Lefschetz Fixed Point Theorem, The Jordan-Brouwer Theorem, The Brouwer Invariance Domain Theorem, Borsuk-Ulam Theorem

Cup product in cohomology.
Ring structure in cohomology, Definition of the cup-product, Example, Relative case, External cup product

Cap product and the Poincar`e duality.
Definition of the cap product, Crash course on manifolds, Poincar´e isomorphism, Some computations

Hopf Invariant
Whitehead product, Hopf invariant

Elementary obstruction theory
Eilenberg-MacLane spaces and cohomology operations, Obstruction theory, Proof of Theorem, Stable Cohomology operations and Steenrod algebra

Cohomology of some Lie groups and Stiefel manifolds
Stiefel manifolds

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