Find a simplicial map from k to k so that the induced map

Question

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.