Geometric topology

For other uses, seeGeometric topology (disambiguation).

ASeifert surfacebounded by a set ofBorromean rings;these surfaces can be used as tools in geometric topology

Inmathematics,geometric topologyis the study ofmanifoldsandmapsbetween them, particularlyembeddingsof one manifold into another.

History

Geometric topology as an area distinct fromalgebraic topologymay be said to have originated in the 1935 classification oflens spacesbyReidemeister torsion,which required distinguishing spaces that arehomotopy equivalentbut nothomeomorphic.This was the origin ofsimplehomotopytheory. The use of the term geometric topology to describe these seems to have originated rather recently.

Differences between low-dimensional and high-dimensional topology

Manifolds differ radically in behavior in high and low dimension.

High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings incodimension3 and above.Low-dimensional topologyis concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.

Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such asexotic differentiable structures onR⁴.Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of thegeneralized Poincaré conjecture;seeGluck twists.

The distinction is becausesurgery theoryworks in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work. Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.

TheWhitney trickrequires 2+1 dimensions, hence surgery theory requires 5 dimensions.

The precise reason for the difference at dimension 5 is because theWhitney embedding theorem,the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via ahomotopyof a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale'sh-cobordism theorem,which works in dimension 5 and above, and forms the basis for surgery theory.

A modification of the Whitney trick can work in 4 dimensions, and is calledCasson handles– because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ( "tower" ) of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.

Important tools in geometric topology

Main article:List of geometric topology topics

Fundamental group

Main article:Fundamental group

In all dimensions, thefundamental groupof a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in dimension 4 and above everyfinitely presented groupis the fundamental group of a manifold (note that it is sufficient to show this for 4- and 5-dimensional manifolds, and then to take products with spheres to get higher ones).

Orientability

Main article:Orientability

A manifold is orientable if it has a consistent choice oforientation,and aconnectedorientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods ofhomology theory,whereas fordifferentiable manifoldsmore structure is present, allowing a formulation in terms ofdifferential forms.An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (afiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

Handle decompositions

Main article:Handle decomposition

A 3-ball with three 1-handles attached.

Ahandle decompositionof anm-manifoldMis a union

$\emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M$

where each$M_{i}$is obtained from$M_{i-1}$ by the attaching of$i$-handles.A handle decomposition is to a manifold what aCW-decompositionis to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world ofsmooth manifolds.Thus ani-handle is the smooth analogue of ani-cell. Handle decompositions of manifolds arise naturally viaMorse theory.The modification of handle structures is closely linked toCerf theory.

Local flatness

Main article:Local flatness

Local flatnessis a property of asubmanifoldin atopological manifoldof largerdimension.In thecategoryof topological manifolds, locally flat submanifolds play a role similar to that ofembedded submanifoldsin the category ofsmooth manifolds.

Suppose addimensional manifoldNis embedded into anndimensional manifoldM(whered<n). If$x\in N,$we sayNislocally flatatxif there is a neighborhood$U\subset M$ofxsuch that thetopological pair$(U,U\cap N)$ishomeomorphicto the pair$(\mathbb {R} ^{n},\mathbb {R} ^{d})$,with a standard inclusion of$\mathbb {R} ^{d}$as a subspace of$\mathbb {R} ^{n}$.That is, there exists a homeomorphism$U\to R^{n}$such that theimageof$U\cap N$coincides with$\mathbb {R} ^{d}$.

Schönflies theorems

Main article:Jordan-Schönflies theorem

The generalizedSchoenflies theoremstates that, if an (n− 1)-dimensionalsphereSis embedded into then-dimensional sphereSⁿin alocally flatway (that is, the embedding extends to that of a thickened sphere), then the pair (Sⁿ,S) is homeomorphic to the pair (Sⁿ,Sⁿ⁻¹), whereSⁿ⁻¹is the equator of then-sphere. Brown and Mazur received theVeblen Prizefor their independent proofsof this theorem.

Branches of geometric topology

Low-dimensional topology

Main article:Low-dimensional topology

Low-dimensional topologyincludes:

• Surfaces(2-manifolds)
• 3-manifolds
• 4-manifolds

each have their own theory, where there are some connections.

Low-dimensional topology is strongly geometric, as reflected in theuniformization theoremin 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and thegeometrization conjecture(now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

2-dimensional topology can be studied ascomplex geometryin one variable (Riemann surfacesare complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.

Knot theory

Main article:Knot theory

Knot theoryis the study ofmathematical knots.While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is anembeddingof acirclein 3-dimensionalEuclidean space,R³(since we're using topology, a circle isn't bound to the classical geometric concept, but to all of itshomeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation ofR³upon itself (known as anambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in otherthree-dimensional spacesand objects other than circles can be used; seeknot (mathematics).Higher-dimensional knots aren-dimensional spheresinm-dimensional Euclidean space.

High-dimensional geometric topology

In high-dimensional topology,characteristic classesare a basic invariant, andsurgery theoryis a key theory.

Acharacteristic classis a way of associating to eachprincipal bundleon atopological spaceXacohomologyclass ofX.The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possessessectionsor not. In other words, characteristic classes are globalinvariantswhich measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts inalgebraic topology,differential geometryandalgebraic geometry.

Surgery theoryis a collection of techniques used to produce onemanifoldfrom another in a 'controlled' way, introduced byMilnor(1961). Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with,handlebody decompositions.It is a major tool in the study and classification of manifolds of dimension greater than 3.

More technically, the idea is to start with a well-understood manifoldMand perform surgery on it to produce a manifoldM′ having some desired property, in such a way that the effects on thehomology,homotopy groups,or other interesting invariants of the manifold are known.

The classification ofexotic spheresbyKervaireandMilnor(1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

See also

• Category:Maps of manifolds
• List of geometric topology topics
• Plumbing (mathematics)

References

• R. B. Sher andR. J. Daverman(2002),Handbook of Geometric Topology,North-Holland.ISBN0-444-82432-4.