2.3 Chain maps and induced homomorphisms

Chain maps are the backbone of homological algebra, connecting different chain complexes. They preserve the structure of chain complexes and allow us to compare them. Understanding chain maps is crucial for grasping how information flows between different algebraic structures.

Induced homomorphisms on homology groups are a key consequence of chain maps. They let us relate the homology of different chain complexes, giving us a powerful tool for studying topological and algebraic properties. This connection is fundamental for many applications in algebra and topology.

Chain Maps and Induced Homomorphisms

Definition and Properties of Chain Maps

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• Chain map $f: C_\bullet \to D_\bullet$ between chain complexes $C_\bullet$ and $D_\bullet$ consists of a family of homomorphisms $f_n: C_n \to D_n$ for each $n \in \mathbb{Z}$ that commute with the boundary operators
  • Commuting with boundary operators means $f_{n-1} \circ \partial^C_n = \partial^D_n \circ f_n$ for all $n \in \mathbb{Z}$, where $\partial^C_n$ and $\partial^D_n$ are the boundary operators of $C_\bullet$ and $D_\bullet$ respectively
  • Commutative diagram illustrates the relationship between chain maps and boundary operators, with vertical arrows representing $f_n$ and horizontal arrows representing boundary operators
• Composition of chain maps $f: C_\bullet \to D_\bullet$ and $g: D_\bullet \to E_\bullet$ defined by $(g \circ f)_n = g_n \circ f_n$ for each $n \in \mathbb{Z}$, resulting in a chain map $g \circ f: C_\bullet \to E_\bullet$
• Identity chain map $\text{id}_{C_\bullet}: C_\bullet \to C_\bullet$ defined by $(\text{id}_{C_\bullet})_n = \text{id}_{C_n}$ for each $n \in \mathbb{Z}$, serving as the identity element for composition of chain maps

Induced Homomorphisms on Homology Groups

• Chain map $f: C_\bullet \to D_\bullet$ induces homomorphisms $H_n(f): H_n(C_\bullet) \to H_n(D_\bullet)$ on homology groups for each $n \in \mathbb{Z}$
  • Induced homomorphism $H_n(f)$ maps homology class $[c] \in H_n(C_\bullet)$ to homology class $[f_n(c)] \in H_n(D_\bullet)$
  • Well-defined since chain maps preserve cycles and boundaries, ensuring that homologous cycles in $C_\bullet$ are mapped to homologous cycles in $D_\bullet$
• Functoriality of induced homomorphisms states that $H_n(g \circ f) = H_n(g) \circ H_n(f)$ for chain maps $f: C_\bullet \to D_\bullet$ and $g: D_\bullet \to E_\bullet$, and $H_n(\text{id}_{C_\bullet}) = \text{id}_{H_n(C_\bullet)}$ for the identity chain map
  • Functoriality allows the study of homology groups and their relationships through the lens of category theory, treating chain complexes as objects and chain maps as morphisms

Naturality of Induced Homomorphisms

• Naturality of induced homomorphisms expresses the compatibility between chain maps and the connecting homomorphisms in long exact sequences of homology groups
  • Given a short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ and chain maps $f_A: A_\bullet \to A'_\bullet$, $f_B: B_\bullet \to B'_\bullet$, and $f_C: C_\bullet \to C'_\bullet$ that commute with the maps in the short exact sequence, the induced homomorphisms on homology groups commute with the connecting homomorphisms in the corresponding long exact sequence of homology groups
• Naturality is a powerful tool for studying the behavior of homology groups under various constructions and transformations in homological algebra
  • Allows the transfer of information between different chain complexes and their homology groups, facilitating the computation and comparison of homology groups in various contexts

Chain Homotopy Equivalence and Quasi-Isomorphisms

Chain Homotopy and Chain Homotopy Equivalence

• Chain homotopy between chain maps $f, g: C_\bullet \to D_\bullet$ is a family of homomorphisms $h_n: C_n \to D_{n+1}$ for each $n \in \mathbb{Z}$ satisfying $f_n - g_n = \partial^D_{n+1} \circ h_n + h_{n-1} \circ \partial^C_n$
  • Chain homotopy measures the "difference" between chain maps $f$ and $g$, with the homomorphisms $h_n$ providing a way to "deform" one chain map into the other
  • Existence of a chain homotopy between $f$ and $g$ implies that they induce the same homomorphisms on homology groups, i.e., $H_n(f) = H_n(g)$ for all $n \in \mathbb{Z}$
• Chain homotopy equivalence between chain complexes $C_\bullet$ and $D_\bullet$ consists of chain maps $f: C_\bullet \to D_\bullet$ and $g: D_\bullet \to C_\bullet$ such that $g \circ f$ is chain homotopic to $\text{id}_{C_\bullet}$ and $f \circ g$ is chain homotopic to $\text{id}_{D_\bullet}$
  • Chain homotopy equivalence is a weaker notion than isomorphism of chain complexes, as it allows for non-invertible chain maps that still induce isomorphisms on homology groups
  • Chain complexes that are chain homotopy equivalent have isomorphic homology groups, i.e., $H_n(C_\bullet) \cong H_n(D_\bullet)$ for all $n \in \mathbb{Z}$

Quasi-Isomorphisms and Their Properties

• Quasi-isomorphism is a chain map $f: C_\bullet \to D_\bullet$ that induces isomorphisms $H_n(f): H_n(C_\bullet) \to H_n(D_\bullet)$ on homology groups for all $n \in \mathbb{Z}$
  • Quasi-isomorphisms are a stronger notion than chain homotopy equivalences, as they directly induce isomorphisms on homology groups without the need for a chain homotopy inverse
  • Composition of quasi-isomorphisms is again a quasi-isomorphism, allowing for the study of chain complexes and their homology groups up to quasi-isomorphism
• Quasi-isomorphisms play a central role in the theory of derived categories and derived functors, where they are used to localize categories of chain complexes with respect to quasi-isomorphisms
  • Localization process allows for the construction of derived categories, where quasi-isomorphic chain complexes are identified and morphisms are formally inverted, leading to a powerful framework for studying homological invariants and constructions
• Quasi-isomorphisms between chain complexes of modules over a ring $R$ induce isomorphisms between the corresponding derived functors, such as Ext and Tor, computed using these chain complexes
  • Allows for the computation and comparison of derived functors using simpler or more convenient chain complex resolutions, providing a valuable tool in homological algebra