Namespaces
Variants
Actions

Difference between revisions of "Homology of a complex"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (MR/ZBL numbers added)
Line 2: Line 2:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. MacLane,   "Homology" , Springer (1963)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR></table>
  
  
Line 10: Line 10:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.H. Spanier,   "Algebraic topology" , McGraw-Hill (1966)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR></table>

Revision as of 21:53, 30 March 2012

The starting point for various homological constructions. Let be an Abelian category and let be a chain complex in , i.e. a family of objects in and morphisms such that for all . The quotient object is called the -th homology of the complex and is denoted by . The family is also denoted by . The concept of the homology of a complex serves as the base for a number of important constructions in homological algebra, commutative algebra, algebraic geometry, and topology. Thus, in topology, each topological space defines a chain complex in the category of Abelian groups: . Here is the group of -dimensional singular chains of , while is the boundary homomorphism. The -th homology of this complex is said to be the -th singular homology group of and is denoted by . The concept of the cohomology of a cochain complex is defined in a dual manner.

References

[1] S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009


Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303
How to Cite This Entry:
Homology of a complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_of_a_complex&oldid=23860
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article