# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

**Tutorials**

Monday, 11:15 - 13:00, Ziskind, Rm 155

## Field of Study, Course Type and Credit Points

Physical Sciences: Lecture; Elective; 2.00 points

Life Sciences (Brain Sciences: Systems, Computational and Cognitive Neuroscience Track): Lecture; Elective; 2.00 points

## Comments

## Prerequisites

Knowledge of group theory and topology and linear algebra.

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**Final assignment**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

**Advanced Topology- continuation from Basic Topology:**

**Lecture 12:**

πi(Sn); i ≤ n. [FF, 9], [HAT, 4.2].

**Lecture 13:**

CW complexes: de?nition, cellular approximation, CW aproximation, Whitehead theorem, computation of π1 and of homologies of CW complexes, obstacles to computation of πn of CW complexes. [GH, 21], [FF, 3], [HAT, 0, 4.1].

__Lecture 14-15:__

(a) Simplicial sets. De?nition, realisation, Kan condition. combinatorial description of homotopy classes of maps between realisations of Kan simplicial sets.

(b) long exact sequence of (Serre) ?bration. Examples. [FF, 7,8], [HAT, 4.2].

(c) Eilenberg-MacLane spaces [FF, 2], [HAT, 4.2].

**Lecture 16-17:**

(a) Relative homotopy groups and long exact sequence a pair. [FF, 8], [HAT, 4.1].

(b) Excision and corolaries: Hurewicz theorem, Freudenthal suspension theorem, stable homotopy groups [FF, 9], [HAT, 4.2].

4.4. Advanced Homology theory.

**Lecture 18-19:**

(a) Kunneth theorem. [GH, 29], [HAT, 3.2,3.B].

(b) Universal coe?cient theorem [GH, 29], [FF, 15], [HAT, 3.1, 3.A]

(c) Cohomology: de?nition, cup product, duality to homologies. [GH, 23, 24], [FF, 14], [HAT, 3.1].

(d) Cohomology with compact support and Borel-Moore homology. [GH, 26], [HAT, 3.3].

**Lecture 20-21:**

Cech (co-)homology. [HAT, 3.3].

**Lecture 22-23:**

(a) Orientation and Poincare duality [GH, 22, 26], [HAT, 3.3].

(b) Relation to Eilenberg-MacLane spaces [FF, 2], [HAT, 4.3]

4.5. Advanced topics.

**Lecture 24-25:**

(a) Sheaf cohomology.

(b) Spectral sequences.

(c) the stable homotopy category and spectra.

(d) Alexander duality

(e) Cohomology operations

(f) Bott periodicity theorem

(g) K-theory

(h) Bordisms

## Learning Outcomes

Upon successful completion of this course students should be able to:

Understand the basic properties of homotopy and horology groups, and will have a glance into more advance topics of Algebraic topology.

## Reading List

TBA