Inmathematics,geometric topology is the study ofmanifolds andmaps between them, particularlyembeddings of one manifold into another.

Geometric topology as an area distinct fromalgebraic topology may be said to have originated in the 1935 classification oflens spaces byReidemeister torsion, which required distinguishing spaces that arehomotopy equivalent but nothomeomorphic. This was the origin ofsimple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently.^[1]

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 incodimension 3 and above.Low-dimensional topology is 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 theory works 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.

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 ahomotopy of 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.

In all dimensions, thefundamental group of 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 group is 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).

A manifold is orientable if it has a consistent choice oforientation, and aconnected orientable 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 manifolds more 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.

Ahandle decomposition of anm-manifoldM is a union

${\displaystyle \mathrm{\varnothing }=M_{-1}\subset M_0\subset M_1\subset M_2\subset \cdots \subset M_{m-1}\subset M_m=M}$

where each${\displaystyle M_i}$ is obtained from${\displaystyle M_{i-1}}$by the attaching of${\displaystyle i}$-handles. A handle decomposition is to a manifold what aCW-decomposition is 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 is a property of asubmanifold in atopological manifold of largerdimension. In thecategory of topological manifolds, locally flat submanifolds play a role similar to that ofembedded submanifolds in the category ofsmooth manifolds.

Suppose ad dimensional manifoldN is embedded into ann dimensional manifoldM (whered <n). If${\displaystyle x\in N,}$ we sayN islocally flat atx if there is a neighborhood${\displaystyle U\subset M}$ ofx such that thetopological pair${\displaystyle (U,U\cap N)}$ ishomeomorphic to the pair${\displaystyle ({\mathbb{R}}^n,{\mathbb{R}}^d)}$, with a standard inclusion of${\displaystyle {\mathbb{R}}^d}$ as a subspace of${\displaystyle {\mathbb{R}}^n}$. That is, there exists a homeomorphism${\displaystyle U\to R^n}$ such that theimage of${\displaystyle U\cap N}$ coincides with${\displaystyle {\mathbb{R}}^d}$.

The generalizedSchoenflies theorem states that, if an (n − 1)-dimensionalsphereS is embedded into then-dimensional sphereSⁿ in alocally flat way (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⁻¹ is the equator of then-sphere. Brown and Mazur received theVeblen Prize for their independent proofs^[2][3] of this theorem.

Low-dimensional topology includes:

• 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 theorem in 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 geometry in one variable (Riemann surfaces are 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 is 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 anembedding of acircle in 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 ofR³ upon 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 spaces and objects other than circles can be used; seeknot (mathematics). Higher-dimensional knots aren-dimensional spheres inm-dimensional Euclidean space.

In high-dimensional topology,characteristic classes are a basic invariant, andsurgery theory is a key theory.

Acharacteristic class is a way of associating to eachprincipal bundle on atopological spaceX acohomology class ofX. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possessessections or not. In other words, characteristic classes are globalinvariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts inalgebraic topology,differential geometry andalgebraic geometry.

Surgery theory is a collection of techniques used to produce onemanifold from 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 manifoldM and 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 spheres byKervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

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

• R. B. Sher andR. J. Daverman (2002),Handbook of Geometric Topology, North-Holland.ISBN 0-444-82432-4.

• Geometric topology
• Geometry processing

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