Differential topology

A branch of topology dealing with the topological problems of the theory of differentiable manifolds and differentiable mappings, in particular diffeomorphisms, imbeddings and bundles. Attempts at a successive construction of topology on the basis of manifolds, mappings and differential forms date back to the end of 19th century (H. Poincaré), but at that time a full realization of this procedure proved impossible. A systematic construction of differential topology could be realized only in the 1930s, as a result of joint efforts of prominent mathematicians.

Important general mathematical concepts were developed in differential topology. These include fibrations and bundles over a space, and the differential-geometric and topological concepts connected with it: connections, $G$-structures, characteristic classes, and also rigged (framed) manifolds. Of major importance in the development of differential topology was the theory of (co)bordisms, with its several applications in algebraic and analytical geometry (the Riemann–Roch theorem), the theory of elliptic operators (the index theorem), and also in topology itself. The 1950s witnessed the discovery of various different smooth structures on spheres; this was followed by the classification of manifolds having the homotopy type of the spheres and by the proof of the generalized Poincaré conjecture: The solution was found for the problem of finding a complete system of diffeomorphism invariants of all simply-connected manifolds (of a dimension not less than 5). In the 1960s fundamental topological problems were solved by applying the methods of differential topology: It was found that the characteristic classes of real manifolds were topologically invariant; the relation between the categories of differentiable, piecewise-linear and topological manifolds were clarified; the methods of classification of smooth manifolds were generalized to include non-simply-connected manifolds (though admittedly not very effectively); and algebraic $K$-theory and Hermitian $K$-theory were created. For non-simply-connected manifolds, fundamental relationships were discovered between the characteristic classes and Hermitian forms over the fundamental group of the manifold and the homology. Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and the theory of Hermitian forms over cochains with an involution. Of major importance are methods concerning the problem of classification of imbeddings of one manifold into another and its various generalizations.

A separate branch of differential topology, related to the calculus of variations, is the global theory of extremals of various functionals on manifolds of geodesics. It strongly influenced the development of topology itself by making possible a classification of vector bundles and, subsequently, by producing a method of studying the topological invariants, provided by $K$-theory. Multi-dimensional global problems of the calculus of variations on manifolds proved to be more difficult; the problem which was principally studied was the problem of minimal surfaces, as the extremals of Dirichlet-type functionals. In the 1970s the theory of elementary particles gave rise to several essentially new problems in multi-dimensional variational calculus.

Another special trend in differential topology, related to differential geometry and to the theory of dynamical systems, is the theory of foliations (Pfaffian systems which are locally totally integrable). Thus, the existence was established of a closed leaf in any two-dimensional smooth foliation on many three-dimensional manifolds (e.g. spheres). Several results in the qualitative theory of foliations were intended to clarify the problem of the existence of an important class of hyperbolic dynamical systems on manifolds. In a subsequent — differential-geometric — development of the theory of foliations, a special analogue of the theory of characteristic classes of fibre bundles, based on the homology theory of Lie algebras of vector fields, was established.

A major achievement of the methods of differential topology is the theory of "generic" singularities of mappings and functions. The methods of differential topology found application in classical problems of algebraic geometry; formerly, results obtained in this field (in particular, theorems on ovals of algebraic curves) were isolated and remained apart from the main development of topology and geometry. The development of differential topology produced several new problems and methods in algebra, e.g., so-called stable algebra, the method of formal groups, etc., and also in the theory of partial differential equations and dynamical systems, functional analysis and geometry.

The interest on the part of modern physics in the methods of differential topology greatly increased in the 1970s. This was due to the increased importance of so-called gauge fields (connections in fibre bundles with space-time as the base) in the theory of elementary particles, the difficult topology of solutions in the theory of liquid crystals, and the theory of phase transitions — particularly, liquid helium at low temperatures.

References

[1]	L.S. Pontryagin, "Smooth manifolds and their applications in homotopy theory" , Moscow (1976) (In Russian) MR0445517 Zbl 0084.19002
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[3]	R. Thom, , Fibre spaces and their applications , Moscow (1958) pp. 293–351 (In Russian; translated from English)
[4]	S.P. Novikov, Izv. Akad. Nauk SSR Ser. Mat. , 28 : 2 (1964) pp. 365–474
[5]	J.W. Milnor, "Differential topology" , Lectures on modern mathematics , II , Wiley (1964) pp. 165–183 MR0178474 Zbl 0142.40803 Zbl 0123.16201
[6]	S. Smale, "A survey of some recent developments in differential topology" Bull. Amer. Math. Soc. , 69 (1963) pp. 131–145 MR0144351 Zbl 0133.16507
[7]	J. Milnor, A. Wallace, "Differential topology" , Moscow (1972) (In Russian; translated from English) MR0353326 Zbl 1225.01040 Zbl 1122.01020 Zbl 0142.40803 Zbl 0123.16201
[8]	J.R. Munkres, "Elementary differential topology" , Princeton Univ. Press (1963) MR0163320 Zbl 0107.17201
[9]	J. Milnor, "Lectures on the $h$-cobordism theorem" , Princeton Univ. Press (1965) MR190942
[10]	V.A. Rokhlin, D.B. Fuks, "Beginners course in topology: Geometric chapters" , Springer (1984) (Translated from Russian) MR759162
[11]	C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972) MR0350744 Zbl 0254.57010
[12]	C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970) MR0431216 Zbl 0219.57024
[13]	F. Browder, "Surgery on simply-connected manifolds" , Springer (1972) MR0358813 Zbl 0239.57016
[14]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[15]	R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[16]	S.P. Novikov, "The topology of foliations" Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 248–278 (In Russian) MR0200938
[17]	, Integer flows and minimal surfaces , Moscow (1973) (In Russian; translated from English and Italian)
[18]	A.S. Mishchenko, "Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis" Russian Math. Surveys , 31 : 2 (1976) pp. 71–138 Uspekhi Mat. Nauk. , 31 : 2 (1976) pp. 69–134 MR2810191 MR2551881 MR2407859 MR2342865 MR2276911 MR2171671 MR1865951 MR1865950 MR1841760 MR1401584 MR1209904 MR1189162 MR1181028 MR1143142 MR1261408 MR0979315 MR0979309 MR0865932 MR0831650 MR0836571
[19]	, Smooth dynamical systems , Moscow (1977) (In Russian; translated from English) Zbl 0357.58006

References

[a1]	M.W. Hirsch, "Differential topology" , Springer (1976) MR0448362 Zbl 0356.57001
[a2]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401