geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated i ...

, a field within mathematics, the

obstruction Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway o ...

to a finitely dominated space ''X'' being

homotopy-equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to a finite

CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...

is its Wall finiteness obstruction ''w(X)'' which is an element in the reduced zeroth

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

\widetilde_0(\mathbb

pi_1(X) The number (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number appears in many formulas across mathematics and physics. It is an irrat ...

of the integral

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...

\mathbb

/math>. It is named after the mathematician

C. T. C. Wall Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where ...

. By work of

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

on finitely dominated spaces, no generality is lost in letting ''X'' be a CW-complex. A ''finite domination'' of ''X'' is a finite CW-complex ''K'' together with maps

r:K\to X

and

i\colon X\to K

such that

r\circ i\simeq 1_X

. By a construction due to Milnor it is possible to extend ''r'' to a homotopy equivalence

\bar\colon \bar\to X

where

\bar

is a CW-complex obtained from ''K'' by attaching cells to kill the relative homotopy groups

\pi_n(r)

. The space

\bar

will be ''finite'' if all relative homotopy groups are finitely generated. Wall showed that this will be the case if and only if his finiteness obstruction vanishes. More precisely, using covering space theory and the

Hurewicz theorem In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...

one can identify

\pi_n(r)

with

H_n(\widetilde,\widetilde)

. Wall then showed that the cellular chain complex

C_*(\widetilde)

is chain-homotopy equivalent to a chain complex

A_*

of finite type of projective

\mathbb

/math>-modules, and that

H_n(\widetilde,\widetilde)\cong H_n(A_*)

will be finitely generated if and only if these modules are stably-free. Stably-free modules vanish in reduced K-theory. This motivates the definition :

in\widetilde_0(\mathbb

References

*. *. *{{citation , last = Rosenberg , first = Jonathan , authorlink = Jonathan Rosenberg (mathematician) , editor1-last = Friedlander , editor1-first = Eric M. , editor1-link = Eric Friedlander , editor2-last = Grayson , editor2-first = Daniel R. , contribution = ''K''-theory and geometric topology , doi = 10.1007/978-3-540-27855-9_12 , location = Berlin , mr = 2181830 , pages = 577–610 , publisher = Springer , title = Handbook of ''K''-Theory , url = http://www.math.uiuc.edu/K-theory/handbook/2-0577-0610.pdf , year = 2005, isbn = 978-3-540-23019-9 Geometric topology Algebraic K-theory Surgery theory

See also

References