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Algebraic Topology - MAT00083M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Eli Hawkins
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Module summary

Algebraic topology relates the shape of topological spaces to the structure of groups. It allows mathematicians to describe the internal structure of spaces, and to tell when two spaces are fundamentally different, using algebra. This module focusses on homology theory, which is one of the crown jewels of 20th century mathematics and provides a paradigm from which substantial parts of modern mathematics have since been developed.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

For MSc students: a first course in topology; a first course in group theory is required.

Module will run

Occurrence Teaching period
A Spring Term 2022-23

Module aims

Algebraic topology relates the shape of topological spaces to the structure of groups. It allows mathematicians to describe the internal structure of spaces, and to tell when two spaces are fundamentally different, using algebra. This module focusses on homology theory, which is one of the crown jewels of 20th century mathematics and provides a paradigm from which substantial parts of modern mathematics have since been developed.

Module learning outcomes

Subject content

·Know how both simplicial and singular homology assign an abelian group to a (nice enough) topological space

  • Know how maps between spaces relate singular homology groups
  • Know how singular homology groups behave when spaces are cut up into pieces

Academic and graduate skills

  • Further development of analytic abilities and logical reasoning; the ability to synthesise elementary ideas from earlier years into a more complex setting where several tools must be used in a new context.

Module content

Syllabus

  • Delta-complexes
  • Simplicial homology
  • Singular homology
  • Homotopy invariance
  • Exact sequences and excision
  • Statement of equivalence of simplicial and singular homologies
  • Applications

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Algebraic Topology		 2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Algebraic Topology		 2 hours 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

A Hatcher, Algebraic Topology, CUP 2001 (S 8.3 HAT)


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.