COURSE SUMMARY
Course Title:
Homological Algebra
Course Code:
18MAT734
Year Taught:
2019
Semester:
2
Type:
Elective
Degree:
Doctoral Programs
School:
School of Arts and Sciences
Campus:
Kochi

'Homological Algebra' is an elective course offered in the second semester of M.Phil. in Mathematics program (Full Time/Part Time) offered by School of Arts & Sciences, Kochi.

#### Syllabus

Unit 1: Simplical Complexes and Homology Groups

Definition and elementary properties-Siplices, Simplical complexes and Simplical maps, homology Groups, Zero-dimensional Homology, The Homology of a cone, Homomorphism induced by Simpicial Maps, Chain Complexes and Acyclic Carriers.

Unit 2: Relative Homology and Eilenberg-Steenrod Axioms

Relative Homology, the Exact Homology Sequence, the Zig-zag Lemma, Mayer-Vietoris Sequences, the Eilenberg-Steenrod Axioms, the Axioms for Simplicial Theory. Categories and functors.

Unit 3: Singular Homology

The Singular Homology Groups, the Axioms for Singular Theory, Excision in Singular Homology, Mayer-Vietoris Sequences, the Isomorphism between Simplical and Singular Homology, More on Quotient Spaces, CW Complexes and their Homology.

Unit 4: Cohomology and Homological Algebra​​​​​​​

The Hom functor, Simplical Cohomology Groups, Relative Cohomology, Cohomology theory, The Ext Functor, The Universal Coefficient Theorem for Cohomology, Torsion Products, The Universal Coefficient Theorem for Homology, Kunneth Theorems (for cohomology and homology proofs can be omitted).

Unit 5: Applications​​​​​​​

Local Homology Groups and Manifolds, the Jordan Curve Theorem, Projective spaces and Lens Spaces, The Cohomology Ring of a Product Space.

#### References

1. Algebraic Topology, Tammo tom Dieck, European Mathematical Society (2008).
2. Algebraic Topology, C.R.F.Maunder, Cambridge University Press, (1980).