Definition Let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over some topological space X. We say that the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic if there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ with the property that p₁ = p₂ ◦ h.  

We can apply Theorem 1.19 in order to derive a criterion for determining whether or not two covering maps over some connected locally path connected topological space are isomorphic.

Theorem 1.21 Let X be a topological space which is both connected and locally path-connected, let X˜₁ and X˜₂ be connected topological spaces, and let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over X. Let w₁ and w₂ be points of X˜₁ and X˜₂ respectively for which p₁(w₁) = p₂(w₂). Then there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ satisfying p₂ ◦ h = p₁ and h(w₁) = w₂ if and only if the subgroups p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) coincide.

Proof Suppose that there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ for which p₂ ◦h = p₁ and h(w₁) = w₂. Thenh_#(π₁(X˜₁, w₁)) = π₁(X˜₂, w₂), and therefore

p_1#(π₁(X˜₁, w₁)) = p_2# (h_#(π₁(X˜₁, w₁))­­) = p_2#(π₁(X˜₂, w₂)).

Conversely suppose that p_1#(π₁(X˜₁, w₁)) = p_2#(π₁(X˜₂, w₂)). It follows from Proposition 1.16 that the covering spaces X˜₁ and X˜₂ are both locally path-connected, since X is a locally path-connected topological space. But X˜₁ and X˜₂ are also connected. It follows from Theorem 1.19 that there exist unique continuous maps h: X˜₁ → X˜₂ and k: X˜₂ → X˜₁ for which p₂ ◦ h = p₁, p₁ ◦ k = p₂, h(w₁) = w₂ and k(w₂) = w₁. But then p₁ ◦ k ◦ h = p₁ and  (k ◦ h)(w₁) = w₁. It follows from this that the composition map k ◦ h is the identity map of X˜₁ (since a straightforward application of Theorem 1.19 shows that any continuous map j: X˜₁ → X˜₁ which satisfies p₁ ◦ j = p₁ and j(w₁) = w₁ must be the identity map of X˜₁). Similarly the composition map h◦k is the identity map of X˜₂. Thus h: X˜₁ → X˜₂ is a homeomorphism whose inverse is k. Moreover p₂ ◦ h = p₂. Thus h: X˜₁ → X˜₂ is a homeomorphism with the required properties.

Corollary 1.22 Let X be a topological space which is both connected and locally path-connected, let X˜₁ and X˜₂ be connected topological spaces, and let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over X. Let w1 and w₂ be points of X˜₁ and X˜₂ respectively for which p₁(w₁) = p₂(w₂). Then the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic if and only if the subgroups p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) are conjugate.

Proof Suppose that the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic. Let h: X˜₁ → X˜₂ be a homeomorphism for which p₂ ◦ h = p₁. Then p_1#(π₁(X˜₁, w₁)) = p_2#(π₁(X˜₂, h(w₁))).

It follows immediately from Lemma 1.7 that the subgroups p_1#(π₁(X˜₁, w₁))  and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) are conjugate.

Conversely, suppose that the subgroups

                      p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂))

of π₁(X, p₁(w₁)) are conjugate. The covering space X˜₂ is both locally path connected (Proposition 1.16) and connected, and is therefore path-connected  (Corollary 1.17). It follows from Lemma 4.7 that there exists a point w of X˜₂ for which p₂(w) = p₂(w₂) = p₁(w₁) and

p_2#(π₁(X˜₂, w)) = p_1#(π₁(X˜₁, w₁)).

اخر الاخبار

اخبار العتبة العباسية المقدسة

المجمع العلمي يحتفي بذكرى ولادة الإمامين الباقر والهادي (عليهما السلام)

قسم الشؤون الفكرية يطلق فعاليات الأسبوع الثالث من برنامج البذرة الصالحة

تكريم الفائزين في المسابقة الثقافية الخاصّة بذكرى ولادة الإمامين الباقر والهادي (عليهما السلام)

بذكرى ولادتهما.. العتبة العباسية المقدسة تستذكر السيرة العطرة للإمامين الباقر والهادي (عليهما السلام)

المزيد

الآخبار الصحية

ثورة طبية.. الذكاء الاصطناعي يكشف سرطان الكلى في وقت قياسي

لتعزيز الطاقة والتغلب على التعب.. عليك بـ7 طرق بسيطة

لتحسين الهضم والتحكم في الشهية.. ما أفضل وقت لتناول التمر؟

خبراء يكشفون تأثير الاستحمام في الظلام على جودة النوم !

المزيد

الآخبار التكنلوجية

إزالة ثاني أكسيد الكربون من الغلاف الجوي.. ما دور المحيطات والبحار؟

نظائر الهيليوم مقياس لاكتشاف رواسب الذهب.. كيف ذلك؟

الصين تطلق أول ناقلة نفط خام ذكية في العالم تعمل بالميثانول

الزمن على المريخ أسرع من الأرض.. دراسة علمية تكشف الفوارق

المزيد

مواضيع ذات صلة

Triple Torus

Thurston Elliptization Conjecture

Thurston,s Geometrization Conjecture

Real Projective Plane

Pseudometric Topology

Pseudocrosscap

Parallelizable

Möbius Strip

Möbius Shorts

Path

Product Topology

Handle