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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. [ 3 ] In practice, there are technical difficulties in using homotopies with certain spaces.
The degree of a map between general manifolds was first defined by Brouwer, [1] who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral). [2]
Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the excision theorem for singular homology and cohomology , there is no simple known way to calculate the homotopy groups of a space by ...
Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It follows that the Euler characteristic is also a homotopy invariant.
The stable homotopy groups of spheres are the direct sum of the image of the J-homomorphism, and the kernel of the Adams e-invariant, a homomorphism from these groups to /. Roughly speaking, the image of the J -homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups.
The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X {\displaystyle X} is denoted by π 1 ( X ) {\displaystyle \pi _{1}(X)} .
Milnor invariants can be considered as invariants of string links, in which case they are universally defined, and the indeterminacy of the Milnor invariant of a link is precisely due to the multiple ways that a given links can be cut into a string link; this allows the classification of links up to link homotopy, as in (Habegger & Lin 1990).
A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork: Let V {\displaystyle V} denote a vector space and V ∞ {\displaystyle V^{\infty }} its one-point compactification , i.e. V ≅ R k {\displaystyle V\cong \mathbb {R} ^{k}} and