# Topology from the differentiable viewpoint

@inproceedings{Milnor1965TopologyFT, title={Topology from the differentiable viewpoint}, author={John W. 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…

## Figures from this paper

## 1,358 Citations

THE HOPF DEGREE THEOREM

- Mathematics
- 2011

This paper develops the theory of differential topology, the study of manifolds and smooth maps between manifolds, towards investigating the classification of homotopy classes of smooth maps to the…

On the homotopy classification of proper Fredholm maps into a Hilbert space

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2020

Abstract We classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our…

Additional structures on inverse images of regular values of smooth maps induced from the source manifolds and cycles of the Reeb spaces

- Mathematics
- 2018

Morse functions and higher dimensional versions and their singularity theory are useful in studying geometric properties of smooth manifolds. In these studies, Reeb spaces, which are defined as the…

Computations in A1-homotopy theory

- Mathematics
- 2020

We introduce Koras-Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine A1-contractible 3-folds.…

Cohomotopy sets of 4-manifolds

- Mathematics
- 2012

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The…

Rochlin ’ s theorem on signatures of spin 4-manifolds via algebraic topology

- Mathematics
- 2015

We give the original proof of Rochlin’s famous theorem on signatures of smooth spin 4-manifolds, which uses techniques from algebraic topology. We have attempted to include enough background and…

REGULARIZATION AND MINIMIZATION OF CODIMENSION-ONE HAEFLIGER STRUCTURES

- Mathematics
- 2017

On compact manifolds of dimensions 4 and more, we give a proof of Thurston's existence theorem for foliations of codimension one; that is, they satisfy some h-principle in the sense of Gromov. Our…

The Topological Classification of Stratified Spaces

- Mathematics
- 1995

Part 1 Manifold theory: algebraic K-theory and topology surgery theory spacification and functoriality applications. Part 2 General theory: definitions and examples classification of stratified…

Introduction to Topology

- Mathematics
- 2001

Topological spaces and operations with them Homotopy groups and homotopy equivalence Coverings Cell spaces ($CW$-complexes) Relative homotopy groups and the exact sequence of a pair Fiber bundles…

## References

SHOWING 1-10 OF 25 REFERENCES

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…

Topology of Fibre Bundles

- Mathematics
- 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…

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…

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…

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…

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…

dc, Varieles differentiables

- Paris : Hermann,
- 1955