# Algebraic Topology II (ΓΕ6)

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### General

School | School of Science |
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Academic Unit | Department of Mathematics |

Level of Studies | Graduate |

Course Code | ΓΕ6 |

Semester | 1 |

Course Title | Algebraic Topology II |

Independent Teaching Activities | Lectures (Weekly Teaching Hours: 3, Credits: 7.5) |

Course Type |
Special background. Specialized general knowledge. Skills development in connections with topology geometry and algebra. |

Prerequisite Courses |
ΓΕ5 - Algebraic Topology I |

Language of Instruction and Examinations |
Greek |

Is the Course Offered to Erasmus Students | Yes (in English) |

Course Website (URL) | See eCourse, the Learning Management System maintained by the University of Ioannina. |

### Learning Outcomes

Learning outcomes |
Algebraic Topology begins its creation by H. Poincare in 1900. In its first thirty years the field seemed limited in application in algebraic geometry, but this changed dramatically in 1930 with the creation of differential topology by G. De Rham and E. Cartan and of homotopy theory by W. Hurewicz and H. Hopf. Its influence began to spread to more and more branches as it gradually took on a central role in mathematics. This course is a continuation of the course Algebraic Topology I and aims in studying and managing advanced skills in order to calculate and solve difficult problems in topology-geometry. The key idea is to attach algebraic structures to topological spaces and their maps in such a away the algebra is both invariant under a variety of deformation of spaces and maps, and computable. Our aim is to transform difficult geometric problems to homotopic ones. We also study and develop homotopical tools. We calculate homological modules as well as cohomological rings for important spaces. Homotopical and cohomological sequences are concerned. |
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General Competences |
Search for analysis and synthesis of data and information related with topological and geometrical problems. Working independently and in a Team work. Working in an interdisciplinary environment aiming at production of new research ideas related to the syllabus of the course. |

### Syllabus

Polyhedral, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, Künneth and universal coefficient theorems, Poincare and Alexander duality theorems. Cofibrations, cofiber homotopy equivalence, fibrations, fiber homotopy equivalence, cofiber-fiber sequences, the cellular approximation theorem. Hopf invariant problem, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, Borsuk-Ulam. Classifying spaces Eilenberg-MacLane spaces, Meyer-Vietoris sequences, vector bundles characteristic classes.

### Teaching and Learning Methods - Evaluation

Delivery |
Face-to-face | ||||||||||||||
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Use of Information and Communications Technology | - | ||||||||||||||

Teaching Methods |
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Student Performance Evaluation |
Written Examination, Oral Presentation, tests, written assignments. |

### Attached Bibliography

- Elements of Homotopy Theory, George Whitehead. Springer Verlag .
- Algebraic Topology, A. Hatcher, https://www.math.cornell.edu/~hatcher/AT/
- A Concise Course in Algebraic Topology, J. P. May, https://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf