स्नातकोत्तर प्रथम वर्षGeometry & Topologyनियोजित
Algebraic topology
Singular and cellular homology, CW complexes, exact sequences and Mayer–Vietoris, homotopy invariance, cohomology and cup products, with applications such as invariance of dimension and the Lefschetz fixed-point theorem.
math.geom.algebraic-topology
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
43 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Compute homology of spheres, projective spaces and surfaces using cellular homology and Mayer–Vietoris.
- Prove that homology is a homotopy-invariant functor and deduce that ℝᵐ and ℝⁿ are not homeomorphic for m ≠ n.
- Compute cohomology rings of simple spaces and use cup products to distinguish spaces with the same homology.
परिभाषा
- The singular chain complex of X has as n-chains the formal integer combinations of continuous maps from the standard n-simplex into X.
- A CW complex is a space built by successively attaching cells of increasing dimension.
- Singular cohomology Hⁿ(X; R) is the cohomology of the dual cochain complex with coefficients in R.
प्रमेय
- Hₙ(Sⁿ) ≅ ℤ for n ≥ 1, and the reduced homology of Sⁿ vanishes in all other degrees.
- Mayer–Vietoris gives a long exact sequence relating the homology of a union to that of its pieces.
- ℝᵐ and ℝⁿ are homeomorphic only if m = n.
- Cup products distinguish ℂP² from S² ∨ S⁴, although the two spaces have isomorphic homology groups.
आम भ्रांतियाँ
- Thinking homology determines homotopy type.
- Confusing homology groups with homotopy groups; the higher homotopy groups of spheres are far more complicated.
आगे ले जाता है
वैकल्पिक खोजें
अधिगम पथ
आगे पढ़ने के लिए
- Allen Hatcher, Algebraic Topology (2002)