For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings.
This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism.
Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other.
A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it.
The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.
from the fundamental group π1(X, x0) to the fundamental group π1(Y, y0) as follows: any element of π1(X, x0), represented by a loop f in X based at x0, is mapped to the loop in π1(Y, y0) obtained by composing with h: Here [f] denotes the equivalence class of f under homotopy, as in the definition of the fundamental group.
is a group homomorphism: (where + denotes concatenation of loops, with the first + in X and the second + in Y).
is the homomorphism induced from h. It may also be denoted as π(h).
Indeed, π gives a functor from the category of pointed spaces to the category of groups: it associates the fundamental group π1(X, x0) to each pointed space (X, x0) and it associates the induced homomorphism
to each base-point preserving continuous map h: (X, x0) → (Y, y0).
To prove it satisfies the definition of a functor, one has to further check that it is compatible with composition: for base-point preserving continuous maps h: (X, x0) → (Y, y0) and k: (Y, y0) → (Z, z0), we have: This implies that if h is not only a continuous map but in fact a homeomorphism between X and Y, then the induced homomorphism
More generally, a simply connected space cannot be homeomorphic to a non-simply-connected space; one has a trivial fundamental group and the other does not.
Therefore, the one-point compactification of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the circle).
For example, R2 and R3 have isomorphic fundamental groups but are still not homeomorphic.
Their fundamental groups are isomorphic because each space is simply connected.
However, the two spaces cannot be homeomorphic because deleting a point from R2 leaves a non-simply-connected space but deleting a point from R3 leaves a simply connected space (If we delete a line lying in R3, the space wouldn’t be simply connected any more.
If A is a strong deformation retract of a topological space X, then the inclusion map from A to X induces an isomorphism between fundamental groups (so the fundamental group of X can be described using only loops in the subspace A).
Any homology theory comes with induced homomorphisms.
240–241) [3] Similarly, any cohomology comes induced homomorphisms, though in the opposite direction (from a group associated with Y to a group associated with X).
[3] Generalizations such as cobordism also have induced homomorphisms.
of topological spaces (possibly with some additional structure) such as the category of all topological spaces Top or the category of pointed topological spaces (that is, topological spaces with a distinguished base point), and a functor
of algebraic structures such as the category of groups Grp or of abelian groups Ab which then associates such an algebraic structure to every topological space, then for every morphism
(which is usually a continuous map, possibly preserving some other structure such as the base point) this functor induces an induced morphism
is a category of groups) between the algebraic structures