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Instructions for oral examination is available under Examination.   For those who wish to earn credit for this course, there is now a selection of exercises under Examination. You are expected to hand in readable solutions. Help and assistance will be offered.

Textbook

Allen Hatcher: Algebraic Topology. The book is also freely available from Allen Hatchers web page. Note the links to corrections and additional exercises.

Program

Discussed topics will be marked green as the course progresses.   Date Topics Pages 1 Quotient spaces, the homotopy relation, cell complexes and construction of new spaces. 1-10, 517-525. 2 The homotopy extension property and collapsing subspaces. 11-17 3 The fundamental group, the fundamental group of  S1  and some consequences, general properties. 21-37 4 Amalgamated products of groups and the van Kampen theorem 40-52, 92-93 5 Covering spaces and their classification 56-70 6 Deck transformations, Eilenberg - Mac Lane spaces of type 1, Delta complexes. 70-78, 87-91, 102-104 7 Simplicial and singular homology, homotopy invariance 104-113 8 Exact sequences in homology, relative homology and excision. 113-126 9 Degree of maps, equivalence of simplicial and singular homology, naturality. 127-131, 134-137 10 Cellular homology, some computations, Euler characteristic. 137-148 11 Homology of groups, Mayer-Vietoris sequences, coefficients in homology. 149-155 12 Axioms for homology theories, categories and their classifying spaces, the fundamental group and H1 160-168 13 Classical applications to topology, simplicial approximation and Lefschetz fixed point theorem. 169-184 14 Cohomology of spaces and the universal coefficient theorem. 185-204 15 The cup product, the cohomology ring, the Künneth theorem, spaces with polynomial cohomology. 206-228 16 Poincaré duality. 230-257 17 The universal coefficient theorem for homology, the cross product in homology and cohomology. 261-280 18 H-spaces and their (co)homology, the cohomology of the special orthogonal group SO(n). 281-302 19 The Bockstein homomorphism, inverse limits and Ext-groups. 303-320 20 Transfer in cohomology, higher homotopy groups. 321-326, 337-346 21 Whiteheads theorem and cellular approximation. 346-357 22 Excision in homotopy and the Hurewicz map. 360-375 23 Fiber bundles and fibrations. 375-388 24 Browns representation of cohomology, obstruction theory 393-419 25 Cohomology of fiber bundles 375-388

Schedule

Läsvecka Dag Tid Lokal 2 må 24 jan 13-15 S1   ti 25 10-12 S2 3 ti 1 feb 10-12 S2   to 3 10-12 S1 4 to 10 10-12 S1   fr 11 10-12 S1 5 må 14 13-15 S1   to 17 10-12 S1 6 må 21 13-15 S1   to 24 10-12 S1 7 to 3 mar 10-12 S1   fr 4 10-12 S1 8 ti 8 13-15 S1   fr 11 13-15 S1   ti 15 13-15 S1   to 17 10-12 S1   to 31 10-12 S2   fr 1 apr 10-12 S1 1 ti 5 13-15 S1   to 7 10-12 S1 2 må 11 10-12 S1   to 14 10-12 S1 3 må 18 10-12 S1   to 21 10-12 S1 4 ti 26 10-12 S2   to 28 10-12 S1 5 må 2 maj 13-15 S2   ti 3 10-12 S2 6 må 9 10-12 S1   to 12 10-12 S1 7 må 16 10-12 S1   to 19 10-12 S1   må 23 10-12 S1   to 26 10-12 S1

Examination

At the end of the course there will be individual oral examinations. You are also expected to solve exercises in the book and on some occasions hand them in for evaluation. Passed examination will give you 5 credit points.   The home work to hand in is: Section in the book Excercises   1.1 10, 16, 20 Choose 2 1.2 4, 9, 16, 20 Choose 3 1.3 4, 5, 9, 15, 19, 21 Choose 4 2.1 4, 6, 17, 27, 29 Choose 4 2.2 3, 4, 9, 13, 15, 16, 22, 25, 26 Choose 6 2.C 2, 7   3.1 5, 6, 7, 11   3.2 3, 4, 6, 7, 8, 11 Choose 4 3.3 3, 7, 10, 16, 20, 25     Instructions for oral examination  If you whish to take an oral exam contact Jan Alve to make an appointment.  The oral exam is arranged as follows. The student is assigned, by chance, one of the tasks in the list below and gets one hour to prepare a short exposition (about 30 min) for the examiner. This will then be followed by a discussion of the subject.   Account for the definition and general properties of the fundamental group and possibilities to compute it, the theory of covering spaces and their classification over a fixed base space, the definition and general properties of singular homology, the possibility to compute the homology and Delta- and CW-complexes, when the homology is finitely generated and the Universal coefficient theorem for homology, the definition and general properties of the cohomology (ring) of a space, the Universal coefficient theorem for cohomology and Kunneths formula (weak version), and exemplify Poncaré duality for closed oriented manifolds, H-spaces, their (co)homology and connection with Hopf-algebras, and exemplify the Leray-Hirsch theorem for fibrations

Supplementary notes

Additional literature

Bott and Tu: Differential forms in algebraic topology. Bredon: Geometry and Topology. Dold: Lectures on Algebraic Topology. Greenberg and Harper: Algebraic Topology. Massey: A basic course in algebraic topology. Matousek: Using the Borsuk-Ulam Theorem; Lectures on Topological Methods in Combinatorics and Geometry (Springer 2002). May: A concise course in algebraic topology. Rotman: An introduction to algebraic topology. Spanier: Algebraic topology. Switzer: Algebraic topology - homology and homotopy. Whitehead: Elements of homotopy theory. George K. Francis: A topological Picture Book.

Resources

Hopf Topology Archive Algebraic Topology Discussion List The Knot Plot Site Topology Atlas (General Topology) Maple computes homology (look under Mathematics - Topology)