M8130 Algebraic topology, tutorial 02, 2024
3.10.2024
Exercise 1. Show that CPn is a CW-com,plex.
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M8130 Algebraic topology, tutorial 02, 2024
3.10.2024
Exercise 2. Let A = {0}%c R. Show that (I, A) does not have the HEP, i e the inclusion A ^ I is not a cofibration.
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M8130 Algebraic topology, tutorial 02, 2024
3.10.2024
Exercise 3. From the lecture we know that A := {I n G N} U {0} as a subspace ofR is not a CW-complex. Show that X := / x {0} U A x / is not a CW-complex either.
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M8130 Algebraic topology, tutorial 02, 2024_ 3.10.2024
Exercise 4. Prove that every compact set A in a CW-complex X can have a nonempty intersection with only finitely many cells.
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M8130 Algebraic topology, tutorial 02. 2024
3.10.2024
Exercise 5. Show that the, Hawaiian earring given by
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M8130 Algebraic topology, tutorial 02, 2024_3.10.2024
Exercise 6. Show that for a short exact sequence O-^A-UB-^C—^Oof abelian groups (or more generally modules over a commutative ring) the following are equivalent:
(1) There exists p : B —> A such that pf = id^.
(2) There, exists q . C    B such that gq = idc-
(3) There exist p : B —> A and q : C —> B such that fp + qg idB.
(Another equivalent condition is B = A®C, with (p,g) and f + q being the respective inverse isomorphisms.)
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M8130 Algebraic topology, tutorial 02, 2024__3.10.2024
Exercise 7. Let 0 ^ A B, 4 C* -> 0 k a short exact sequence of chain modules. We have defined the connecting homom,orphism. d* : H„(C) —> Hn-\{A) by the formula d*[c] = [a], where dc = 0, /(a) = db and g(b) — c. Show that this definition does not depend on a nor b.
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