Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title MANIFOLDS, HOMOLOGY AND MORSE THEORY
Code MATH410
Coordinator Dr A Pratoussevitch
Mathematical Sciences
Anna.Pratoussevitch@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 7 FHEQ First Semester 15

Aims

To give an introduction to the topology of manifolds, emphasising the role of homology as an invariant and the role of Morse theory as a visualising and calculational tool.


Learning Outcomes

To be able to:

•   give examples of manifolds, particularly in low dimensions;

•   compute homology groups, Euler characteristics and degrees of maps in simple cases;

•   determine whether an explicitly given function is Morse and to identify its critical points and their indices;

•   use the Morse inequalities to estimate the ranks of homology groups;

•   use the Morse complex to compute Euler characteristics and, in simple cases, homology.


Syllabus

  • Smooth manifolds (embedded in Rn): coordinate charts, tangent spaces, examples, manifolds with boundary.
  • Homology: chains, cycles, boundaries, the definition of homology and simple examples. Functoriality of homology, homotopy equivalences and the homotopy invariance of homology. Exact sequences of Abelian groups. Relative homology and the long exact sequence of a pair (proof non-examinable). Excision (statement only). Computations of homology. Applications to degrees of maps, Euler characteristics and fixed point theorems.
  • Morse theory: smooth maps, non-degenerate critical points, the Hessian and the index. Morse functions, Sard’s lemma and the existence of Morse functions (non-examinable), the Morse lemma. Change in homotopy type at critical values and consequences for homology. The Morse inequalities. Brief treatment of the Morse complex. Applications to Euler characteristics, computations of homology and Poincare duality.

Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101; MATH244; MATH102; MATH103  

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

Programme:G101 Year:3 Programme:G101 Year:4 Programme:MMAS Year:1

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam  2.5 hours  First  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) Final exam rubric: All questions carry equal weight. Full marks can be obtained by fully answering FIVE questions. Only the best FIVE solutions will be counted.  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes