Reference no: EM131103449
Q1. Let n ∈ Z. Find triangulations K and L for S1 and a simplicial map f: |K| → |L| so that
- if α is the sum of the 1-simplices of K, then α generates H1(K),
- if β is the sum of the 1-simplices of L, then β generates H1(L), and
- the map f∗: H1(K) → H1(L) is such that f∗(α) = nβ.
Your map f should, in some sense, be equivalent to a familiar map g: S1 → S1. What is g?
Q2. Consider the complex T below. It is a triangulation for the torus.
Let σ be the sum of all the 2-simplices of T, oriented counter-clockwise. Then σ is a generating cycle for H2(T) ≅ Z. Let α = [a, b] + [b, c] + [c, a] and β = [a, d] + [d, e] + [e, a]. (The orientation on [a, b] is directed from a to b.) Then α and β are generators for H1(T) ≅ Z ⊕ Z.
(a) Define simplicial maps f, g, h, and k from T to itself extending the following data:
(b) Compute the values of f∗, g∗, h∗, and k∗ on σ, α, and β.
Q3. Let T be as above, and let S be the the boundary of a 3-simplex having vertices A, B, C, and D. (So |S| ≅ S2.) Let τ be the 2-cycle ∂[A, B, C, D]. Then τ generates H2(S) ≅ Z.
(a) Let f: T → S be the simplicial map with f(m) = f(r) = A, f(p) = B, f(b) = f(u) = C, and all other vertices are sent to D. Compute the induced homomorphism f∗: H2(T) → H2(S).
(b) Let g: T → S be the simplicial map which agrees with f on all vertices except g(r) = C. Compute g∗.
(c) Let h: T → S be the simplicial map which agrees with g on all vertices except h(u) = A. Compute h∗.
(d) What are the induce maps f∗, g∗, and h∗ on H1?
Q4. Alter the schematic for T above by switching the labels for d and e on the right to get a triangulation K for the Klein bottle. Let α and β be as before.
(a) Find a simplicial map from K to K so that the induced map on homology sends α to β.
(b) Show that there is no simplicial map from K to K sending β to α.
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