an induced homomorphism. A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often...

¯ c ∗ ( X , A ; G ) {\displaystyle {\bar {H}}_{c}^{*}(X,A;G)} . Induced homomorphism of this cohomology is defined as the Alexander cohomology theory...

of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function...

definitions. A homomorphism of topological groups means a continuous group homomorphism G → H. Topological groups, together with their homomorphisms, form a...

_{0}(Y)} is bijective, and for every point x in X and every n ≥ 1, the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) {\displaystyle f_{*}\colon...

quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily seen to be well-defined). If A is...

substructures are precisely the induced subgraphs. However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding...

be different.) On the other hand, a module homomorphism M → K is a pure embedding if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K...

Then, for a homomorphism f : G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to...

the (usual) Gysin homomorphism induced by the zero-section embedding X ′ ↪ N {\displaystyle X'\hookrightarrow N} . The homomorphism i! encodes intersection...

properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from G ∗ H to K. Unless one of the...

and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as π ( f ) {\displaystyle \pi (f)} or, more...

pushforward homomorphism f ∗ : H i ( X ) → H i ( Y ) {\displaystyle f_{*}:H_{i}(X)\to H_{i}(Y)} on homology and a pullback homomorphism f ∗ : H i ( Y...

{\displaystyle f\colon X\to Y} and a point x in X, consider for any n ≥ 0 the induced homomorphism f ∗ : π n ( X , x ) → π n ( Y , f ( x ) ) , {\displaystyle f_{*}\colon...

{\displaystyle [k]} is a ring homomorphism and is called the de Rham homomorphism. It is not difficult to show that the de Rham homomorphism is a natural transformation...

characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism ( R , m ) ↪ ( S , n ) {\displaystyle...

∗ ( X ) {\displaystyle f^{*}\colon H^{*}(Y)\to H^{*}(X)} is the induced homomorphism in cohomology, then f ∗ ( α ⌣ β ) = f ∗ ( α ) ⌣ f ∗ ( β ) , {\displaystyle...

{\textstyle d\neq q} with d ∣ q {\textstyle d\mid q} such that the induced homomorphism χ ~ : ( Z / q Z ) × → C ∗ {\textstyle {\widetilde {\chi }}\colon...

{\displaystyle p_{2}\colon E_{2}\to B} with the same base space is a fibration homomorphism if the following diagram commutes: The mapping f {\displaystyle f} is...

continuous homomorphism between real Lie groups turns out to be (real) analytic. The composition of two Lie homomorphisms is again a homomorphism, and the...

{\displaystyle X} ? The answer is yes, provided we also know the induced homomorphisms π 1 ( A ∩ B , ∗ ) → π 1 ( A , ∗ ) {\displaystyle \pi _{1}(A\cap...

isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over...

continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but...

group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object. Homomorphism groups...

and (T, ∨), a homomorphism of (join-) semilattices is a function f: S → T such that f(x ∨ y) = f(x) ∨ f(y). Hence f is just a homomorphism of the two semigroups...

{Gray}\longrightarrow }~\operatorname {coker} c} where d is a homomorphism, known as the connecting homomorphism. Furthermore, if the morphism f is a monomorphism...

of bits Ω = 2 Z , {\displaystyle \Omega =2^{\mathbb {Z} },} the induced homomorphism is the Baker's map. The dyadic sequence is then just the sequence...

arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism group...

given a group homomorphism f : G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is...