Please use question 3 as a reference for Question 5. ONLY ANSWER QUESTION 5 PLEASE

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In class I claimed that P2#P2 # P2 = T#P?. This exercise will walk you through why this fact is true. See Figure 1 on the next page. a) Prove that a gluing diagram for T #P2 is shown in (A). b) Prove that a gluing diagram for P#PP#P2 is shown in (B). c) Prove that the two gluing diagrams are equivalent via cut and paste. I have shown some of the steps. Wherever there is a star and an arrow, there are missing steps that you need to fill in. a) Give an intuitive reason why the connected sum operation does not have an inverse. b) Using Exercise 3, rigorously prove that the connected sum operation does not have an inverse