In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the pushforward of a
continuous function :
between two
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s is a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
between the
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s for
.
Homology is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
which converts a topological space
into a sequence of homology groups
. (Often, the collection of all such groups is referred to using the notation
; this collection has the structure of a
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
.) In any
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
, a functor must induce a corresponding
morphism. The pushforward is the morphism corresponding to the homology functor.
Definition for singular and simplicial homology
We build the pushforward homomorphism as follows (for singular or simplicial homology):
First we have an induced homomorphism between the singular or simplicial
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
and
defined by composing each singular n-
simplex :
with
to obtain a singular n-simplex of
,
:
. Then we extend
linearly via
.
The maps
:
satisfy
where
is the
boundary operator between chain groups, so
defines a
chain map
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
.
We have that
takes cycles to cycles, since
implies
. Also
takes boundaries to boundaries since
.
Hence
induces a homomorphism between the homology groups
for
.
Properties and homotopy invariance
Two basic properties of the push-forward are:
#
for the composition of maps
.
#
where
:
refers to identity function of
and
refers to the identity isomorphism of homology groups.
A main result about the push-forward is the homotopy invariance: if two maps
are homotopic, then they induce the same homomorphism
.
This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:
The maps
induced by a homotopy equivalence
are isomorphisms for all
.
References
*
Allen Hatcher Allen, Allen's or Allens may refer to:
Buildings
* Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee
* Allen Center, a skyscraper complex in downtown Houston, Texas
* Allen Fieldhouse, an indoor sports arena on the Unive ...
''Algebraic topology.''Cambridge University Press, and {{ISBN, 0-521-79540-0
Topology
Homology theory