Let \(X \subset \mathbb{R}^{n+1}\) be the union of the infinite sequence of spheres \(S_k^n\) of radius \(1/k\) and center \((1/k, 0, \ldots, 0)\). Show that \(\pi_i(X) = 0\) for \(i < n\) and construct a homomorphism from \(\pi_n(X)\) onto \(\prod_k \pi_n(S_k^n)\).
Solution
For each \(k\), let \[ r_k:X\to S_k^n \] collapse every sphere \(S_j^n\) with \(j\ne k\) to the common tangent point. This is continuous because the diameters of the spheres tend to zero. Hence there is a homomorphism \[ \Phi:\pi_n(X)\longrightarrow\prod_{k\ge1}\pi_n(S_k^n), \qquad \Phi([f])=([r_kf])_k. \]
First let \(i<n\). Given \(f:S^i\to X\), compactness of the domain and cellular approximation for this shrinking CW-type union allow \(f\) to be homotoped into the \(i\)-skeleton. Each \(S_k^n\) has no positive cells below dimension \(n\), so this skeleton is just the common basepoint. Therefore \[ \boxed{\pi_i(X)=0\qquad(i<n).} \] Equivalently, one may contract each compact image successively off the \(n\)-spheres since \(S^i\to S^n\) is nullhomotopic for \(i<n\), with continuity at the common point guaranteed by the shrinking diameters.
To prove surjectivity of \(\Phi\), choose arbitrary classes \[ a_k\in\pi_n(S_k^n)\cong\mathbb Z. \] Choose pairwise disjoint closed \(n\)-balls \(B_k\subset S^n\) whose diameters tend to zero and which accumulate only at one point \(q\). Collapse the complement of the interiors of all \(B_k\)’s to the common basepoint of \(X\). On each quotient \[ B_k/\partial B_k\cong S^n \] map to \(S_k^n\) by a representative of degree \(a_k\). Since \(\operatorname{diam}(S_k^n)\to0\), these maps fit continuously at the accumulation point \(q\). The resulting map \[ f:S^n\to X \] satisfies \[ [r_kf]=a_k \] for every \(k\). Thus \[ \boxed{\Phi:\pi_n(X)\twoheadrightarrow\prod_k\pi_n(S_k^n).} \]