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Homotopy theory is the study of topological spaces and homotopy equivalences. These equivalences are weaker than isomorphism. An isomorphism is given by two maps $f : X \leftadj Y : g$, such that the both compositions are equal to identities. A homotopy equivalence weakens this by requiring the compositions to be homotopic to identities. Some properties of spaces, such as some kinds of connectedness, only depend on the homotopy type. Such properties are homotopy invariants. \todo{not happy with this yet...} |
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Homotopy theory is the study of topological spaces and homotopy equivalences. These equivalences are weaker than isomorphism. An isomorphism is given by two maps $f : X \leftadj Y : g$, such that the both compositions are equal to identities. A homotopy equivalence weakens this by requiring the compositions to be homotopic to identities. Some properties of spaces, such as some kinds of connectedness, only depend on the homotopy type. Such properties are homotopy invariants. |
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Examples of homotopy invariants are homology groups $H_n(X)$ and homotopy groups $\pi_n(X)$. The latter is defined as the set of continuous maps $S^n \to X$ up to homotopy. Despite the easy definition, the groups $\pi_n(S^k)$ are very hard to calculate and much of it is even unknown as of today. |
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Examples of homotopy invariants are homology groups $H_n(X)$ and homotopy groups $\pi_n(X)$. The latter is defined as the set of continuous maps $S^n \to X$ up to homotopy. Despite the easy definition, the groups $\pi_n(S^k)$ are very hard to calculate and much of it is even unknown as of today. |
Joshua Moerman
10 years ago
