Mapping cone of a degree d circle map

The map $h([t])=e^{2\pi it}$ identifies the quotient circle with the unit circle: it is a continuous bijection, and inverse angular charts are continuous on arcs, including arcs crossing the quotient seam. Give it the counterclockwise orientation. Then $h{f}_d{h}^{-1}(z)=z^d$. For d>0 the fibre over 1 is $e^{2\pi ik\mathrm{/}d}$, 0≤k<d. In positive angular coordinates at each preimage and at 1, the map is u↦du. Interpolating the positive coefficient d to 1 on a sufficiently small arc gives a homotopy of punctured pairs. Thus F3 identifies its local multiplier with that of an orientation-preserving circle rotation, namely +1 by the identity and rotation homotopy, or the singleton-fibre case of F4. For d<0 the same calculation reduces to angular reflection, whose degree is −1 by the circle clause of F5. Hence F4 gives degree d for every nonzero d. For d=0 the map is constant and F5 gives degree zero.

For the fundamental group take U to be the target circle together with cone heights s<2/3, and V to be cone heights s>1/3 including its tip. They are open in the quotient, their union is the cone space, and their intersection is a circle times (1/3,2/3). U retracts to the target by decreasing height, V contracts to the tip by increasing height, and the intersection retracts to its circle. All are path-connected. Choose the basepoint at source [0], height 1/2 and transport to the target vertex along the height segment. The overlap generator maps to a^d in π1(U) by F9 and trivially in π1(V). Thus F8 gives the presentation $⟨a\mid a^d=1⟩=\mathbb{Z}\mathrm{/}d\mathbb{Z}$. This calculation does not infer H1 from π1.

The cone on the source circle is the disk via $[z,s]\mapsto (1-s)z$; its boundary at s=0 attaches by f_d. Thus the mapping cone has one vertex, one loop edge and one 2-cell. F2 and step 1.1 identify its attaching coefficient in F6 as d, so its cellular complex is $0\to \mathbb{Z}\stackrel{d}{\to }\mathbb{Z}\stackrel{0}{\to }\mathbb{Z}\to 0$. Direct kernels and images give the stated H0,H1,H2 and vanishing above dimension two, and F7 identifies them with singular homology.

When d=0 the cellular map is zero, so H1=H2=Z and the group relation is empty. When d=±1 the cellular map is an isomorphism and the group relation kills a, giving the claimed vanishings. For instance d=−2 gives H1=π1=Z/2Z and H2=0. No inference of contractibility from these invariants is made.