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Algebraic Topology

Lefschetz Fixed Point Theorem

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The Lefschetz Fixed Point Theorem applies only to continuous maps from a topological space to itself with a finite number of fixed points.

False. The Lefschetz Fixed Point Theorem can apply even if there are an infinite number of fixed points, or none, because it deals with the existence of fixed points indirectly through Lefschetz numbers.

For a continuous map $f: X \rightarrow X$ on a compact oriented manifold, if the Lefschetz number $L(f)$ is nonzero, then $f$ must have at least one fixed point.

True. The Lefschetz Fixed Point Theorem states that if the Lefschetz number $L(f)$ is nonzero, then $f$ has at least one fixed point.

The Lefschetz Fixed Point Theorem cannot be applied to spaces that are not compact.

False. While the classic version of the theorem is stated for compact spaces, there are generalizations that apply to non-compact spaces as well.

The Lefschetz number $L(f)$ is simply the number of fixed points of a map $f$ .

False. The Lefschetz number is defined algebraically in terms of traces of induced maps on homology groups, not just as a count of fixed points.

If a map $f: X \rightarrow X$ has a Lefschetz number of zero, $L(f) = 0$ , it implies that $f$ has no fixed points.

False. A Lefschetz number of zero does not guarantee the absence of fixed points; $f$ can still have fixed points, their contributions might just cancel out in the Lefschetz number calculation.

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