Geometry II

The course aims to provide students with further topics in general topology, focusing in particular on paracompact spaces and topological manifolds, and to introduce them to algebraic topology, with particular emphasis on homotopy and the fundamental group.

Knowledge and understanding

Students should be able to understand the statements and proofs of fundamental theorems in topology, both from a theoretical perspective (developing a rigorous mathematical language and assimilating definitions, theorems, and the main ideas underlying their proofs) and from a practical perspective (solving problems).

Making judgements

Students should acquire a well-developed degree of independent judgement with regard to the evaluation and interpretation of solutions to problems in topology; be able to construct and develop logical arguments, with a clear identification of assumptions and conclusions; and be able to recognize correct proofs and identify flawed reasoning.

These objectives will be pursued through problem-solving activities during the course and through supplementary support for the course. These activities will provide students with opportunities to develop their decision-making and judgement skills independently. The above-mentioned skills will be developed through interactive teaching: students enrolled in the Mathematics degree programme will constantly assess their own knowledge by working independently or collaboratively in small groups on simple new problems proposed during problem-solving sessions and support sessions.

Communication skills

Students should be able to communicate information, ideas, problems, solutions, and conclusions clearly and unambiguously; present, orally or in writing, the most important theorems of algebraic topology in a clear and comprehensible manner; and work in groups while operating with defined degrees of autonomy.

The final examination will also provide students with a further opportunity to deepen and assess their abilities in analyzing, developing, and communicating their work.

Learning skills

Students should have developed the skills necessary to undertake further studies with a high degree of autonomy; possess the learning skills and a high standard of knowledge and competence necessary to attend lectures and undertake the coursework of Master's degree programmes in Mathematics; and have a flexible mindset, enabling them to adapt readily to new problems and to integrate effectively into professional environments.

Learning skills will be developed throughout the degree programme through the distribution of the overall workload, which assigns appropriate and significant weight to independent study.

Applying knowledge and understanding

Students should be able to prove known mathematical results using techniques different from those previously learned; construct rigorous proofs; and construct simple examples.

The above-mentioned skills will be developed through interactive teaching: students will constantly assess their own knowledge by working independently or collaboratively in small groups on simple new problems proposed during both regular problem-solving sessions and supplementary support sessions.

If the course is delivered in blended or distance-learning mode, the necessary adjustments may be introduced to the arrangements described above in order to ensure that the planned syllabus, as set out in the course description, is properly covered.

Geometria 2– 12 credits (94 total hours)

Course organization – Second semester: 6 credits – 47 hours

• 35 hours of lectures
• 12 hours of problem-solving sessions

Lectures and in-class problem solving. During the course, the instructor will provide sets of exercises accompanied by appropriate hints to support students' preparation. Students are strongly encouraged to work through these exercise sets and write out their solutions in detail. Students may also work on the exercise sets in groups, provided that each student writes up their own solution independently. Part of the course will be devoted to discussing these exercises, and students will be invited to present their solutions at the board.

Part of the syllabus (up to a maximum of 3 credits) may be taught by a visiting foreign or Italian professor with relevant expertise.

1. Set-theoretic preliminaries: Zermelo's well-ordering theorem and the order topology on the first uncountable ordinal.
2. Fundamentals of paracompact spaces. Partitions of unity.
3. Products of paracompact spaces, Michael's Line.
4. The Theorems of Michael and Stone.
5. Introduction to topological manifolds.
6. Homotopy of continuous functions and topological spaces.
7. Paths and homotopy of paths.
8. Introduction to the fundamental group.
9. An outline of covering spaces. The fundamental group of the circle.
10. Applications of the fundamental group of the circle (the Brouwer Fixed Point Theorem and the Borsuk-Ulam theorem in dimension 2. The Fundamental Theorem of Algebra).
11. The fundamental group of the sphere.

• Course notes.
• Marco Manetti, Topologia, editore Springer, 2014.

The final examination for Geometry 2 consists of a written examination lasting approximately three hours and an oral examination, and may be taken at the end of the course. Students who receive a grade below 15/30 on the written examination are not admitted to the oral examination and must retake the written examination.

A midterm examination will be held during the break between the first and second semesters, on the same dates as the regular examination sessions. The midterm examination consists of a written test covering the course content taught during the first semester. Students who pass the midterm examination with a grade of at least 15/30 may take the oral examination during the June/July examination sessions.

The assessment of learning may also be conducted remotely if circumstances so require. In such cases, the duration of the written examination may be subject to change.

As a general rule, grades will be assigned according to the following criteria:

• Fail: The student has not acquired the basic concepts and is unable to solve the exercises.

• 18–23/30: The student demonstrates a minimal understanding of the basic concepts; their ability to present and connect the course material is limited, but they are able to solve simple exercises.

• 24–27/30: The student demonstrates a good understanding of the course material; their ability to present and connect the material is good, and they can solve exercises with only a few errors.

• 28–30/30, with honours: The student has acquired all the course material and is able to present it comprehensively and make connections between the different topics with a critical perspective; they solve exercises completely and without errors.

To take the final examination, students must register through the SmartEdu portal*. For any technical issues concerning examination registration, students should contact the Academic Office*.