1. Geometric Topology Arxiv
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4. Topology Books Pdf
5. Topology Mathematics

Geometric Topology Easter 2018 1 Introduction 2 2 Wall’s Finiteness Obstruction 2 3 The Whitehead torsion 7 4 The s-cobordism theorem 11 5 Siebenmann’s end theorem 22 6 Fibering over a circle 30. 1Introduction The basic idea is that we have some sort of object, and we want to know when it is.

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of a geometric object that are preserved under continuousdeformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

1. In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology, concerning in particular objects of three or four dimensions.
2. Abstract: This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

• 3Concepts
• 4Topics
• 5Applications
• 7References

Motivation[edit]

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory.

Similarly, the hairy ball theorem of algebraic topology says that 'one cannot comb the hair flat on a hairy ball without creating a cowlick.' This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.

To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.^[1]

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from 'squishing' some larger object.

An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends on the font used, and on whether the strokes making up the letters have some thickness or are ideal curves with no thickness. The figures here use the sans-serifMyriad font and are assumed to consist of ideal curves without thickness. Homotopy equivalence is a coarser relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the 'hole' part.

Homeomorphism classes are:

• no holes corresponding with C, G, I, J, L, M, N, S, U, V, W, and Z;
• no holes and three tails corresponding with E, F, T, and Y;
• no holes and four tails corresponding with X;
• one hole and no tail corresponding with D and O;
• one hole and one tail corresponding with P and Q;
• one hole and two tails corresponding with A and R;
• two holes and no tail corresponding with B; and
• a bar with four tails corresponding with H and K; the 'bar' on the K is almost too short to see.

Homotopy classes are larger, because the tails can be squished down to a point. They are:

• one hole,
• two holes, and
• no holes.

To classify the letters correctly, we must show that two letters in the same class are equivalent and two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.

Letter topology has practical relevance in stenciltypography. For instance, Braggadocio font stencils are made of one connected piece of material.

History[edit]

Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.^[2] Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.^[2] On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signalling the birth of topology.^[3][4]

Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.^[5] Listing introduced the term 'Topologie' in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.^[6] The English form 'topology' was used in 1883 in Listing's obituary in the journal Nature to distinguish 'qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated'.^[7] The term 'topologist' in the sense of a specialist in topology was used in 1905 in the magazine Spectator.^{[citation needed]}

Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.^[5]

Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906.^[8] A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term 'topological space' and gave the definition for what is now called a Hausdorff space.^[9] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.^[10]

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology.

Concepts[edit]

Topologies on sets[edit]

The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology tells how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

Geometric Topology Arxiv

1. Both the empty set and X are elements of τ.
2. Any union of elements of τ is an element of τ.
3. Any intersection of finitely many elements of τ is an element of τ.

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X_τ may be used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called a neighborhood of x.

Continuous functions and homeomorphisms[edit]

A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.

Manifolds[edit]

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds).

Topics[edit]

General topology[edit]

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.^[11][12] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.

Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.

Algebraic topology[edit]

Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.^[13] The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

The most important of these invariants are homotopy groups, homology, and cohomology.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Differential topology[edit]

Differential topology is the field dealing with differentiable functions on differentiable manifolds.^[14] It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly 'flatten out' certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Geometric topology[edit]

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.^[15][16] Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem.

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

Low-dimensional topology is strongly geometric, as reflected in the uniformization 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, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.

2-dimensional topology can be studied as complex 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.

Generalizations[edit]

Occasionally, one needs to use the tools of topology but a 'set of points' is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory,^[17] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.^[18]

Applications[edit]

Biology[edit]

Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.^[19] Topology is also used in evolutionary biology to represent the relationship between phenotype and genotype.^[20] Phenotypic forms that appear quite different can be separated by only a few mutations depending on how genetic changes map to phenotypic changes during development. In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns of activity in neural networks.

Computer science[edit]

Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is:

1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
2. Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.^[21]
3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.^[21]

Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.^[22]

Physics[edit]

Topology is relevant to physics in areas such as condensed matter physics,^[23]quantum field theory and physical cosmology.

May 15, 2018 (7,108). Free nexus guitar expansion.

The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials.^[24] The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.^[25] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

The topological classification of Calabi-Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.^[26]

In cosmology, topology can be used to describe the overall shape of the universe.^[27] This area of research is commonly known as spacetime topology.

Robotics[edit]

The possible positions of a robot can be described by a manifold called configuration space.^[28] In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose.^[29]

Games and puzzles[edit]

Tanglement puzzles are based on topological aspects of the puzzle's shapes and components.^{[30][31][32][33]}

Fiber art[edit]

In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.^[34]

See also[edit]

References[edit]

Citations[edit]

1. ^Hubbard, John H.; West, Beverly H. (1995). Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems. Texts in Applied Mathematics. 18. Springer. p. 204. ISBN978-0-387-94377-0.
2. ^ ^abCroom 1989, p. 7
3. ^Richeson 2008, p. 63
4. ^Aleksandrov 1969, p. 204
5. ^ ^abcRicheson (2008)
6. ^Listing, Johann Benedict, 'Vorstudien zur Topologie', Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
7. ^Tait, Peter Guthrie, 'Johann Benedict Listing (obituary)', Nature 27, 1 February 1883, pp. 316–317
8. ^Fréchet, Maurice (1906). Sur quelques points du calcul fonctionnel. PhD dissertation. OCLC8897542.
9. ^Hausdorff, Felix, 'Grundzüge der Mengenlehre', Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
10. ^Croom 1989, p. 129
11. ^Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
12. ^Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.
13. ^Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. ISBN0-521-79160-X, 0-521-79540-0.
14. ^Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN978-0-387-95448-6.
15. ^Budney, Ryan (2011). 'What is geometric topology?'. mathoverflow.net. Retrieved 29 December 2013.
16. ^R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. ISBN0-444-82432-4
17. ^Johnstone, Peter T. (1983). 'The point of pointless topology'. Bulletin of the American Mathematical Society. 8 (1): 41–53. doi:10.1090/s0273-0979-1983-15080-2.
18. ^Artin, Michael (1962). Grothendieck topologies. Cambridge, MA: Harvard University, Dept. of Mathematics. Zbl0208.48701.
19. ^Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society. ISBN978-0-8218-3678-1
20. ^Stadler, Bärbel M.R.; Stadler, Peter F.; Wagner, Günter P.; Fontana, Walter (2001). 'The Topology of the Possible: Formal Spaces Underlying Patterns of Evolutionary Change'. Journal of Theoretical Biology. 213 (2): 241–274. CiteSeerX10.1.1.63.7808. doi:10.1006/jtbi.2001.2423. PMID11894994.
21. ^ ^abGunnar Carlsson (April 2009). 'Topology and data'(PDF). Bulletin (New Series) of the American Mathematical Society. 46 (2): 255–308. doi:10.1090/S0273-0979-09-01249-X.
22. ^Vickers, Steve (1996). Topology via Logic. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press. ISBN9780521576512.
23. ^'The Nobel Prize in Physics 2016'. Nobel Foundation. 4 October 2016. Retrieved 12 October 2016.
24. ^Stephenson, C.; et., al. (2017). 'Topological properties of a self-assembled electrical network via ab initio calculation'. Sci. Rep. 7: 41621. Bibcode:2017NatSR..741621S. doi:10.1038/srep41621. PMC5290745. PMID28155863.
25. ^Cambou, Anne Dominique; Narayanan, Menon (2011). 'Three-dimensional structure of a sheet crumpled into a ball'. Proceedings of the National Academy of Sciences. 108 (36): 14741–14745. arXiv:1203.5826. Bibcode:2011PNAS.10814741C. doi:10.1073/pnas.1019192108. PMC3169141. PMID21873249.
26. ^Yau, S. & Nadis, S.; The Shape of Inner Space, Basic Books, 2010.
27. ^The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ISBN0-8247-7437-X)
28. ^John J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004
29. ^Michael Farber, Invitation to Topological Robotics, European Mathematical Society, 2008
30. ^https://math.stackexchange.com How to reason about disentanglement puzzles.
31. ^Horak, Mathew (2006). 'Disentangling Topological Puzzles by Using Knot Theory'. Mathematics Magazine. 79 (5): 368–375. doi:10.2307/27642974. JSTOR27642974..
32. ^http://sma.epfl.ch/Notes.pdf A Topological Puzzle, Inta Bertuccioni, December 2003.
33. ^https://www.futilitycloset.com/the-figure-8-puzzle The Figure Eight Puzzle, Science and Math, June 2012.
34. ^Eckman, Edie: Connect the shapes crochet motifs: creative techniques for joining motifs of all shapes. ©2012 Storey Publishing

Bibliography[edit]

• Aleksandrov, P.S. (1969) [1956], 'Chapter XVIII Topology', in Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A. (eds.), Mathematics / Its Content, Methods and Meaning (2nd ed.), The M.I.T. Press
• Croom, Fred H. (1989), Principles of Topology, Saunders College Publishing, ISBN978-0-03-029804-2
• Richeson, D. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press

Further reading[edit]

• Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, ISBN3-88538-006-4.
• Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
• Breitenberger, E. (2006). 'Johann Benedict Listing'. In James, I.M. (ed.). History of Topology. North Holland. ISBN978-0-444-82375-5.
• Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN978-0-387-90125-1.
• Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN978-1-4196-2722-4. (Provides a well motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, covering spaces, and orbit spaces.)
• Wacław Sierpiński, General Topology, Dover Publications, 2000, ISBN0-486-41148-6
• Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press. ISBN978-1-56025-826-1. (Provides a popular introduction to topology and geometry)
• Gemignani, Michael C. (1990) [1967], Elementary Topology (2nd ed.), Dover Publications Inc., ISBN978-0-486-66522-1

External links[edit]

• Hazewinkel, Michiel, ed. (2001) [1994], 'Topology, general', Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN978-1-55608-010-4
• Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.
• Topology at Curlie
• The Topological Zoo at The Geometry Center.
• Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.
• Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

Geometric Topology Pdf File

• 3Important tools in geometric topology
• 4Branches of geometric topology

History[edit]

Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently.^[1]

Differences between low-dimensional and high-dimensional topology[edit]

Manifolds differ radically in behavior in high and low dimension.

Algebraic And Geometric Topology Pdf

High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 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 as exotic differentiable structures on R⁴. Thus the topological classification of 4-manifolds is in principle easy, 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 the generalized Poincaré conjecture; see Gluck twists.

The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.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 the Whitney 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 a homotopy 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's h-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 called Casson 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[edit]

Fundamental group[edit]

In all dimensions, the fundamental 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 every finitely 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).

Orientability[edit]

A manifold is orientable if it has a consistent choice of orientation, and a connected 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 of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential 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 (a fiber 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[edit]

A handle decomposition of an m-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 a CW-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 of smooth manifolds. Thus an i-handle is the smooth analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory. The modification of handle structures is closely linked to Cerf theory.

Local flatness[edit]

Local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds.

Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If ${\displaystyle x\in N,}$ we say N is locally flat at x if there is a neighborhood ${\displaystyle U\subset M}$ of x such that the topological pair${\displaystyle (U,U\cap N)}$ is homeomorphic 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 the image of ${\displaystyle U\cap N}$ coincides with ${\displaystyle {\mathbb{R}}^d}$.

Schönflies theorems[edit]

The generalized Schoenflies theorem states that, if an (n − 1)-dimensional sphereS is embedded into the n-dimensional sphere Sⁿ in a locally 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), where Sⁿ⁻¹ is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their independent proofs^[2][3] of this theorem.

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Branches of geometric topology[edit]

Low-dimensional topology[edit]

Low-dimensional topology includes:

• Surface (topology)s (2-manifolds)

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

Low-dimensional topology is strongly geometric, as reflected in the uniformization 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 the geometrization 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 as complex 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[edit]

Knot theory is the study of mathematical 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 an embedding of a circle in 3-dimensional Euclidean space, R³ (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R³ upon itself (known as an ambient 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 other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

High-dimensional geometric topology[edit]

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

A characteristic class is a way of associating to each principal bundle on a topological spaceX a cohomology class of X. The cohomology class measures the extent to which the bundle is 'twisted' — particularly, whether it possesses sections or not. In other words, characteristic classes are global invariants which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry.

Surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by Milnor (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 manifold M and perform surgery on it to produce a manifold M ′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other interesting invariants of the manifold are known.

The classification of exotic spheres by Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

See also[edit]

References[edit]

Topology Books Pdf

1. ^https://math.meta.stackexchange.com/questions/2840/what-is-geometric-topology Retrieved May 30, 2018
2. ^Brown, Morton (1960), A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc., vol. 66, pp. 74–76. MR0117695
3. ^Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 1959 59–65. MR0117693

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

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