Algebraic topology

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• ConceptHomotopy

  A homotopy between two continuous maps $f, g \colon X \to Y$ is a continuous deformation of $f$ into $g$, parametrised by a time $t \in [0, 1]$. Formally it is a continuous map $H \colon X \times [0, 1] \to Y$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$. Homotopy is the basic notion of algebraic topology. Homotopic maps induce the same maps on homology and cohomology, and homotopies that fix a base point give the same maps on homotopy groups.

• ConceptHomotopic maps

  Two continuous maps $f, g \colon X \to Y$ are homotopic, written $f \simeq g$, when there is a homotopy from $f$ to $g$. Being homotopic is an equivalence relation on the set of continuous maps from $X$ to $Y$. Its equivalence classes are the homotopy classes, and the set of them is written $[X, Y]$. Algebraic topology studies spaces through such sets of homotopy classes.

• ConceptRelative homotopy

  A homotopy relative to a subspace $A$ is a homotopy that keeps every point of $A$ fixed during the whole deformation. It lets one deform maps while keeping a boundary condition, such as the end points of a path. Path homotopy, the fundamental group and deformation retractions are all defined with relative homotopies.

• ConceptHomotopy equivalence

  A homotopy equivalence is a continuous map $f \colon X \to Y$ that has an inverse up to homotopy. This is a map $g \colon Y \to X$ with $g \circ f$ homotopic to $\id_X$ and $f \circ g$ homotopic to $\id_Y$. It is a weaker notion than a homeomorphism. Homotopy invariants, such as the fundamental group and homology, cannot distinguish spaces that are related by a homotopy equivalence.

• ConceptHomotopy type

  Two topological spaces have the same homotopy type when there is a homotopy equivalence between them. The homotopy type of a space is its class under this equivalence relation. Most invariants of algebraic topology, such as homotopy groups, homology and cohomology, depend only on the homotopy type. Classifying spaces up to homotopy type is a central aim of homotopy theory.

• ConceptContractible space

  A contractible space is a topological space that has the homotopy type of a point. Equivalently, it is nonempty and its identity map can be deformed continuously into a constant map. All homotopy groups and all reduced homology groups of a contractible space vanish. Contractible spaces are the trivial objects of homotopy theory.