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The cup product provides a way of combining cohomology classes of a topological space. It gives the cohomology ring its ring structure, and is used to study the intersection properties of the manifold.

Betti numbers are integers that give an important inductive count of the maximum number of cuts that can be made without dividing a space into two pieces, indicating the rank of the homology groups.

The Lefschetz Fixed Point Theorem provides a criterion for determining when a continuous map from a compact space to itself has a fixed point, based on the trace of the induced map on homology.

The fundamental group is the set of loop classes in a topological space, based at a point, under the operation of concatenation. It is used to study the 1-dimensional hole structure of spaces and is a basic invariant in algebraic topology.

A fiber bundle in algebraic topology is a space that is locally a product space, but globally may have a different topological structure. It is used to generalize the notions of 'vector bundles' and 'covering spaces'.

Homology is a sequence of abelian groups or vector spaces, which are invariants of a topological space that provide a global way to discern the shape of a space. It is used to classify spaces by the cycles that are not boundaries.

Higher homotopy groups are generalizations of the fundamental group that measure the higher-dimensional 'holes' of a topological space. They are important invariants in homotopy theory.

The Euler characteristic is an integer that is an invariant of a finite dimensional complex, giving a rough measure of its shape or structure. It is defined by the alternating sum of the numbers of cells in each dimension.

Cohomology is a sequence of abelian groups or vector spaces that represent contravariant functors from the category of topological spaces to the category of graded abelian groups. It characterizes the ability to distinguish one topological space from another.

Covering spaces are spaces that 'cover' another space in a specific way that is locally similar to the base space. Understanding covering spaces helps in analyzing the fundamental group and is essential in the classification of surfaces.

Poincaré duality is a fundamental theorem in algebraic topology which states that for a closed orientable manifold, there is an isomorphism between the $nth$ cohomology group and the $(d-n)th$ homology group, where $d$ is the dimension of the manifold.