स्नातक (उन्नत)Geometry & Topologyनियोजित
Homotopy and the fundamental group
Homotopy of maps and paths, the fundamental group, covering spaces, the computation π₁(S¹) ≅ ℤ, the Seifert–van Kampen theorem and applications such as the two-dimensional Brouwer fixed-point theorem.
math.geom.fundamental-group
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
40 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Define homotopy and the fundamental group, and prove that the fundamental group is a homotopy invariant.
- Compute π₁(S¹) ≅ ℤ using covering spaces and deduce the Brouwer fixed-point theorem for the disc.
- Use the Seifert–van Kampen theorem to compute fundamental groups of wedges of circles and of surfaces.
परिभाषा
- A homotopy between maps f, g : X → Y is a continuous map H : X × [0, 1] → Y with H(x, 0) = f(x) and H(x, 1) = g(x).
- π₁(X, x₀) is the group of homotopy classes of loops based at x₀, with concatenation as the operation.
- A covering map p : E → B is a continuous surjection such that each point of B has an open neighbourhood U whose preimage is a disjoint union of open sets, each mapped homeomorphically onto U.
प्रमेय
- π₁(S¹) ≅ ℤ.
- Brouwer: every continuous map from the closed disc to itself has a fixed point.
- The fundamental group of a wedge of two circles is the free group on two generators.
- For path-connected, locally path-connected and semilocally simply connected spaces, connected covering spaces correspond to conjugacy classes of subgroups of π₁.
आम भ्रांतियाँ
- Thinking fundamental groups are always abelian.
- Believing spaces with isomorphic fundamental groups are homeomorphic; S² and a point both have trivial fundamental group.
आगे ले जाता है
वैकल्पिक खोजें
- Homotopy type theory In homotopy type theory the circle is a higher inductive type, and π₁(S¹) ≅ ℤ has a synthetic proof.
- ← Functors
अधिगम पथ
आगे पढ़ने के लिए
- Allen Hatcher, Algebraic Topology (2002)
- James R. Munkres, Topology