Algebraic Topology-Ii - BU 2021 Question Paper

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[Maximum Marks: Reg. 85

Pvt. 100

Note :- All questions from each section carry equal marks. All questions are compulsory and answer limit are approximately 250 words. Start the answer of each section from new page. Maximum limit of pages of answer booklet are approximately 16 pages. Answer should be written by the student in his/her own handwriting mandatory. The first page of answersheet should be download by the student from university website is mandatory.

Define free product of the groups.

Let P be a polygonal region; let w = (a₁)^ε₁......(a_n)^ε_n be a labelling scheme for all edges of P. Let x be the resulting quotient space, Let π : P → x be the quotient map. If π maps all the vertices of P to a single point x₀ of x, and if a₁, ----,a_k are the distinct labels that appear in the labelling scheme, then prove that π₁ (x, x₀) is isomorphic to the quotient of the free group on k-generators α₁,---, α_k by the least normal sub group containing the element (α₁)^ε₁......(α_n)^ε_n

Write down the elementary operating that can be performed on a labelling scheme w₁,---,w_m without affecting the resulting quotient space x.

Let P: E → B and P': E' → B be covering maps and let p (e₀) = p'(e'₀)=b₀. then prove that covering maps p and p' are equivalent if and only if the subgroups H₀ = P_*(π₁ (E,e₀)) and H'₀ = p'_*(π₁ (E',e'₀)) of π₁ (B, b₀) are conjugate.

Let x be a linear graph. If C is a compact subspace of x, then show that there exists a finite subgrph y of x that contains C.