Examples of homotopy

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This will imply the homotopy invariance of the real-valued index.
Its elements are circular paths (continuous images of a circle with orientation) identified up to continuous deformations (homotopy).
The homotopy functor seems unsatisfactory for investigating vectors of more than two processes.
Then is a contracting homotopy for the complex of essential chains, so this complex is a resolution.
The paths and the homotopy can be assumed smooth.
Then induces an isomorphism of the first homotopy groups of the underlying surfaces.
The goal will be to find a reasonable representative map in a given homotopy class which somehow carries the minimal dynamical information for that class.
Our goal is to define variation and degeneration operators on homotopy.
Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments.
In algebraic topology, homotopy is a continuous deformation of mappings; in particular, two paths are homotopic if one can be continuously deformed into the other.
Section 6 describes the crucial relationship between homotopy variation and degeneration operators.
Moreover, the equivariant homomorphisms and are unique up to smooth homotopy.
The assertion that and are uniquely defined up to smooth homotopy follows from the more general statement about the lifting of homotopies.
It follows that itself is a covariant homotopy equivalence.
The entropy stability results in this section are based on a homotopy approach.
The marked length spectrum of is the map associating the length of the closed orbit to the homotopy class.

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