Mathematics — Graph Theory · mapping-torus-homotopy-groups
Problem
Let f:Sα2∨Sβ2→Sα2∨Sβ2 be the map which is the identity on the Sα2 summand and which on the Sβ2 summand is the sum of the identity map and a homeomorphism Sβ2→Sα2. Let X be the mapping torus of f, the quotient space of (Sα2∨Sβ2)×I under the identifications (x,0)∼(f(x),1). The mapping torus of the restriction of f to Sα2 forms a subspace A=S1×Sα2⊂X. Show that the maps π2(A)→π2(X)→π2(X,A) form a short exact sequence 0→Z→Z⊕Z→Z→0, and compute the action of π1(A) on these three groups. In particular, show the action of π1(A) is trivial on π2(A) and π2(X,A) but is nontrivial on π2(X).