Theorem 1.3

Every closed trajectory of the system () contains a rest point in its interior.

Algebraic topology is a branch of geometry where properties of space are studied by assigning algebraic invariants (such as groups, rings etc.,) to space. Thus to each topological space $X$ we attach an algebraic object such as a group $h(X)$ and to each continuous map $f : X \longrightarrow Y$ we attach a group homomorphism $h(f) : h(X) \longrightarrow h(Y)$ satisfying two basic properties:

1. If $X, Y$ and $Z$ are three topological spaces and $f : X \longrightarrow Y$ and $g : Y \longrightarrow Z$ are continuous maps, then the corresponding group homomorphisms $h(f) : h(X) \longrightarrow h(Y)$, $h(g) : h(Y) \longrightarrow h(Z)$ and $h(g\circ f) : h(X) \longrightarrow h(Z)$ must satisfy the condition
   $$h(g\circ f) = h(g)\circ h(f).$$

2. The identity map $i : X \longrightarrow X$ corresponds to the identity map $h(i) : h(X) \longrightarrow h(X)$

These properties are summarized by the statement that $h$ is a (covariant) functor from the category of topological spaces to the category of groups. We shall provide formal definitions of a category and functor elucidating them through examples as we go along.

We shall introduce two important functors - the fundamental groups and the homology groups. We also indicate how these functors help in the understanding (under restrictive conditions) of two fundamental problems in topology - the extension problem and the lifting problem. The Tietze's extension theorem which provides a solution to the extension problem in certain special but important cases, is recalled in lecture 3 where we also place it against the general background of the extension problem. The extension problem reappears again in lecture 10 in connection with the Brouwer's fixed point theorem. Certain questions in complex analysis lead us naturally to the lifting problem as elaborated in lecture 18.

The course is organized as follows. Lectures 1 through 26 constitute the first part on fundamental groups and covering spaces. The second part on singular homology is covered in lectures 28 through 40. We begin with a review of general topology in the next four lectures. We shall touch upon some of the important results on compactness, connectedness, path-connectedness and their local analogues. This is followed by a longer chapter on quotient spaces with a large supply of examples that would occur frequently in the subsequent lectures. The exercises at the end of the lectures are designed as a warm up on these notions. The universal properties of the quotient is emphasized. We shall introduce the notion of a topological group in lecture 5 and discuss some important examples.

In the next lecture we introduce one of the principal thespians of the play - the fundamental group of a topological space. The theme will be developed in the subsequent lectures. The first non-trivial result is that the fundamental group of a circle is the group of integers which in turn implies several important results such as the Brouwer's fixed point theorem and the Perron-Frobenius theorem from matrix theory. The theory of covering spaces will be developed in lectures 13-17. The theory of covering spaces is important in many areas of mathematics but we shall study it here in close connection with the theory of the fundamental group. We introduce one of the fundamental problems in topology namely, the lifting problem for which an elegant solution is available in the context of covering spaces.

Many important spaces in mathematics such as the Klein's bottle, projective spaces and Riemann surfaces (the torus being an important example) occur as orbit spaces under the action of discrete groups. Lecture 18 is devoted to many of these examples. Unfortunately limitations in space and time prevent us from discussing the existence of a universal covering for a space.

Algebraic topology is certainly not a stand alone subject and we have tried (to the extent possible) to indicate connections with other areas of mathematics. #4957#>