Unit-I

Geometric Complexes and Polyhedra: Introduction. Examples. Geometric Complexes and Polyhedra; Orientation of geometric complexes.
Simplicial Homology Groups: Chains, cycles, Boundaries and homology groups, Examples of homology groups; The structure of homology groups.

Unit-II

The Euler Poincare’s Theorem; Pseudomanifolds and the homology groups of Sn.[Chapter 1 Sections 1.1 to 1.4 & Chapter 2 Sections 2.1 to 2.5 from the text].

Unit-III

Simplicial Approximation: Introduction; Simplicial approximatin; Induced homomorphisms on the Homology groups; The Brouwer fixed point theorem and related results;

Unit-IV

The Fundamental Group: Introduction; Homotopic Paths and the Fundamental Group; The Covering Homotopy Property for S1;
[Chapter 3 Sectins 3.1 to 3.4; Chapter 4 Sections 4.1 to 4.3]

Unit-V

Examples of Fundamental Groups; The Relation Between H1(K) and p1(iKi); Covering Spaces: The definition and some examples. Basic properties of covering spaces. Classification of covering spaces. Universal covering spaces. Applications.
[Chapter 4: Sections 4.4, 4.5; Chapter 5 Sections 5.1 to 5.5 from the text]

Course Out Comes:

CO 1: To understand the concept complexes define homology groups
CO 2: To obtain homology groups for various pseudo manifolds
CO 3: To prove Brouwer fixed point theorem and understand its uses
CO 4: To familiarise the concept of homotopy theory and its role in topological spaces Co 5: To find out the fundamental groups of various spaces and anayse the topological structures.

Textbooks

1. Fred H. Croom: Basic Concepts of Algebraic Topology, UTM, Springer, NY, 1978.

References

1. Eilenberg S and Steenrod N: Foundations of Algebraic Topology, Princetion Univ. Press, 1952.
2. S.T. Hu: Homology Theory, Holden-Day, 1965.
3. S.T. Hu: Homology Theory, Academic Press, 1959.