Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. Homotopical connectivity is more formally defined based on the homotopy groups of the space. To say that a space is n"-connected (sometimes, n"-simple connected) is to say that its first "n" homotopy groups are trivial. To say that a map is "n"-connected means that it is an isomorphism "up to dimension "n," in homotopy". The concept of "n"-connectedness generalizes the concepts of path -connectedness and simple connectedness. Definition. The homotopical connectivity of a space "X", denoted formula_1, is the largest integer "d" such that X contains no holes with boundary dimension at most "d". Equivalently, it is the largest integer "d" such that all homotopy groups of order at most d are the trivial group. A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer "d" such that X contains a hole with dimension "d". This connectivity parameter is denoted by formula_2, and it differs from the previous parameter by 2: formula_3. Homotopical connectivity of spheres. In general , for every integer "d", formula_13 (and formula_14)"" The proof requires two directions: Section::::"n"-connected space. A topological space "X" is said to be "n"-connected if it has no holes of dimension at most "formula_20.". Equivalently, every sphere with boundary dimension at most "formula_20" in "X" can be shrunk continuously to a single point. Equivalently, every continuous function formula_22, for formula_23 is homotopic to a constant function. In particular: where formula_29 denotes the "i"-th homotopy group and 0 denotes the trivial group. The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected, respectively, which is  useful in defining 0-connected and 1-connected maps, as below. The "0th homotopy set" can be defined as: This is only a pointed set, not a group, unless "X" is itself a topological group; the distinguished point is the class of the trivial map, sending "S" to the base point of "X". Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set. The definition of homotopy groups and this homotopy set require that "X" be pointed (have a chosen base point), which cannot be done if "X  is empty. A topological space X" is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points "x" and "x" in "X" can be connected with a continuous path which starts in "x" and ends in "x", which is equivalent to the assertion that every mapping from "S" (a discrete set of two points) to "X" can be deformed continuously to a constant map. With this definition, we can define "X" to be "n"-connected if and only if Section::::"n"-connected map. The corresponding "relative" notion to the "absolute" notion of an  n-connected "space" is an "n"-connected "map", which is defined as a map whose homotopy fiber "Ff" is an ("n" − 1)-connected space. In terms of homotopy groups, it means that a map formula_32 is "n"-connected if and only if: The last condition is frequently confusing; it is because the vanishing of the ("n" − 1)-st homotopy group of the homotopy fiber "Ff" corresponds to a surjection on the "n" homotopy groups, in the exact sequence: If the group on the right formula_37 vanishes, then the map on the left is a surjection. Low-dimensional examples: "n"-connectivity for spaces can in  turn be defined in terms of "n"-connectivity of maps: a space "X" with basepoint "x" is an "n"-connected space if and only if the inclusion of the basepoint formula_38 is an "n"-connected map. The single point set is contractible, so all its homotopy groups vanish, and thus "isomorphism below "n" and onto at "n"" corresponds to the first "n" homotopy groups of "X" vanishing. Interpretation. This is instructive for a subset: an "n"-connected inclusion formula_39 is one such that, up to dimension "n" − 1, homotopies in the larger space "X" can be homotoped into homotopies in the subset "A . For example, for an inclusion map formula_39 to be 1-connected, it must be: One-to-one on formula_44 means that if there is a path connecting two points formula_45 by passing through X," there is a path in "A" connecting them, while onto formula_46 means that in fact a path in "X" is homotopic to a path in "A." In other words, a function which is an isomorphism on formula_47 only implies that any elements of formula_48 that are homotopic in "X" are "abstractly" homotopic in "A" – the homotopy in "A" may be unrelated to the homotopy in "X  – while being n"-connected (so also onto formula_49) means that (up to dimension "n" − 1) homotopies in "X" can be pushed into homotopies in "A". This gives a more concrete explanation for the utility of the definition of "n"-connectedness: for example, a space where the inclusion of the "k"-skeleton is "n"-connected (for "n" > "k") – such as the inclusion of a point in the "n"-sphere – has the property that any cells in dimensions between "k" and "n" do not affect the lower-dimensional homotopy types. Identities. Let "K" and "L" be non-empty cell complexes. Their join is commonly denoted  by "K"*"L". Then:"" The identity is simpler with the eta notation: Relations with other concepts. Homological connectivity. Hurewicz theorem relates the homotopical connectivity formula_1 to the homological connectivity"," denoted by formula_53. Suppose first that "X" is simply-connected, that is, formula_54. Hurewicz theorem say that there is an isomorphism between the abelization of the first homotopy group formula_46 and the first homology group formula_56. Since formula_57, this implies formula_58. So formula_59 too. Moreover, for any formula_60, the homotopy group formula_29 and the homology group formula_62 are isomorphic. Since formula_63 for any formula_64 and formula_65 when formula_66 , the same is true  for the formula_62. Therefore: formula_68If "X" is not simply-connected (formula_69), then formula_70 still holds, but the inequality may be strict. Homotopy principle. In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions formula_71 into a more general topological space, such as the space of all continuous maps between two associated spaces formula_72 are "n"-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving h-principles.