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Invariant (mathematics)
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=== Independent of presentation === Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the [[Euler characteristic]] of a [[cell complex]] is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying [[topological space]] (the [[manifold]]) β as different cell complexes give the same underlying manifold, one may ask if the function is ''independent'' of choice of ''presentation,'' in which case it is an ''intrinsically'' defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense. The most common examples are: * The [[Differentiable manifold#Definition|presentation of a manifold]] in terms of coordinate charts β invariants must be unchanged under [[change of coordinates]]. * Various [[manifold decomposition]]s, as discussed for Euler characteristic. * Invariants of a [[presentation of a group]].
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