स्नातकोत्तर प्रथम वर्षGeometry & Topologyपाठ उपलब्ध

Differential topology

Smooth manifolds up to diffeomorphism: regular values and Sard's theorem, transversality, the degree of a map (mod 2 and oriented), the Brouwer fixed-point theorem, vector fields and the Poincaré–Hopf theorem, the hairy ball theorem and a glimpse of Morse theory.

math.geom.differential-topology

पाठ खोलें इस विषय तक पहुँचने के लिए आवश्यक सब कुछ दिखाएँ

इस पर आधारित

Smooth manifolds

42 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)

आप सीखेंगे

State Sard's theorem and use regular values to define the degree of a smooth map between compact manifolds.
Prove the Brouwer fixed-point theorem and the hairy ball theorem using degree.
State the Poincaré–Hopf theorem and compute indices of zeros of vector fields.

परिभाषा

A point y is a regular value of a smooth map f : M → N if the derivative of f is surjective at every point of f⁻¹(y).
Two submanifolds are transverse if at each intersection point their tangent spaces together span the tangent space of the ambient manifold.
The degree of a smooth map f : M → N between closed connected oriented manifolds of the same dimension is the signed count of preimages of a regular value.

प्रमेय

Sard: the critical values of a smooth map have measure zero.
The degree is independent of the regular value and is a homotopy invariant.
Hairy ball: every continuous tangent vector field on S²ⁿ vanishes somewhere.
Poincaré–Hopf: the sum of the indices of the isolated zeros of a vector field on a closed manifold equals its Euler characteristic.

आम भ्रांतियाँ

Thinking 'almost every value is regular' means every value is regular; critical values exist, but they form a set of measure zero.
Believing homeomorphic manifolds are always diffeomorphic; Milnor's exotic 7-spheres are homeomorphic but not diffeomorphic to the standard S⁷.

आगे ले जाता है

Low-dimensional topology: 3- and 4-manifolds

वैकल्पिक खोजें

Low-dimensional topology: 3- and 4-manifolds In dimension 4 the smooth and topological worlds part company: ℝ⁴ admits uncountably many pairwise non-diffeomorphic smooth structures, and no other ℝⁿ admits more than one.
Dynamical systems and chaos Morse theory and the Poincaré–Hopf theorem show how the topology of a manifold forces critical points of functions and zeros of flows.
← Dynamical systems and chaos

अधिगम पथ

Geometry and topology

आगे पढ़ने के लिए

John Milnor, Topology from the Differentiable Viewpoint (1965)
Victor Guillemin and Alan Pollack, Differential Topology (1974)