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Abstract: Can one assign an integer $n(X)$ to a topological space $X$ with the following properties: (1) $n(X)$ captures a geometric or topological property of $X$, (2) $n(X)$ is computable, (3) $n(X)$ is a topological invariant, i.e., if $X$ is homeomorphic to $Y$, then $n(X)=n(Y)$? In the first half of the talk we survey several classical ways of doing this. These are (1) dimension theory which deals with the dimension of a space $X$, (2) the Euler characteristic of $X$, (3) the category of $X$ which is the minimum number of contractible spaces needed to construct $X$. In each case we define the invariant, work out examples and present some illustrative theorems. In the second half of the talk we generalize to continuous mappings $f:X\to Y$ of spaces. We discuss two distinct ways to assign an integer to $f$: (1) the Lefschetz number of $f$ which is a generalization of the Euler characteristic and is used to determine if $f$ has a fixed point, (2) the category of $f$ which generalizes the category of a space.
This talk is meant to be elementary and expository.
This talk will be accessible to graduate students.
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