# Topology from the differentiable viewpoint

@inproceedings{Milnor1965TopologyFT, title={Topology from the differentiable viewpoint}, author={J. Milnor}, year={1965} }

Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 of a mapping20Smooth homotopy and smooth isotopy205Oriented manifolds26The Brouwer degree276Vector fields and the Euler number327Framed cobordism the Pontryagin construction42The Hopf theorem508Exercises52AppClassifying 1… Expand

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#### References

SHOWING 1-10 OF 25 REFERENCES

Introduction to Differentiable Manifolds

- Mathematics
- 1964

Foreword.- Acknowledgments.- Differential Calculus.- Manifolds.- Vector Bundles.- Vector Fields and Differential Equations.- Operations on Vector Fields and Differential Forms.- The Theorem of… Expand

Generalized Poincare's Conjecture in Dimensions Greater Than Four

- Mathematics
- 1961

Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called… Expand

Lectures on Differential Geometry

- Mathematics
- 1964

Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations… Expand

Topology of Fibre Bundles

- Mathematics, Physics
- 1951

Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds and… Expand

A survey of some recent developments in differential topology

- Mathematics
- 1963

1. We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there… Expand

A proof of the nonretractibility of a cell onto its boundary

- Mathematics
- 1963

SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is nor- mally no other outlet. A PROOF OF THE… Expand

The Behavior of a Function on Its Critical Set

- Mathematics
- 1939

Referring to the next section for any unfamiliar notation or definition, let us consider the following statement. If m > 0 n > 1, R is an open subset of En , and f is a function on R to El of class… Expand

Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension

- Mathematics
- 1935

Die Frage, fur welche Dimensionszahlen N und n mit N>n es moglich ist, die Sphare S N wesentlich auf die Sphare S n abzubilden1), ist meines Wissens bisher nur in zwei Fallen beantwortet: 1. Fur… Expand