By Hartwig Hirschfeld

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**Example text**

Clearly IF (S 0 ) is a single point. Let B be a cell complex such that S n = |B|, and let β be an n-cell of B. Any element x ∈ IF (S n ) can be represented by a block bundle ξ over B. Let k, l be trivialisations of ξ|β, ξ|B − β, and let h = k−1 l : F × ∂β−−→F × ∂β. Since F is compact, there are finite simplicial complexes K, L with |K| = F × β, |L| = F × (B − β) and such that h is simplicial. Clearly the simplicial isomorphism class of the triple (K, L, h) determines x completely. But there are only countably many such classes (of triples), so IF (S n ) is countable.

Then θ induces isomorphisms θ∗ : πr (GF /P LF ) −−→ πr ((G/P L)F ) for r ≥ 1. Proof. Since the proof is essentially the same as the proof of Theorem 5, we shall not give the details. To prove that θ∗ is surjective, let B, β, ξ, Q, W be as above. Since Q has a boundary ∂Q such that π1 (∂Q)−−→π1 (Q − W ) is an isomorphism, it is unnecessary to cut out a disc from Q. 3 of [17] to construct a manifold E ⊃ W with boundary ∂E and a simple homotopy equivalence ψ : E, ∂E−−→Q, ∂Q with ψ|W equal to the identity.

Similarly i : E(∂ξ|C)−−→p−1 |C| ∩ ∂E is a homotopy equivalence, so i is a homotopy equivalence of pairs. 1 of [5] that (p−1 |C|, p−1 |C| ∩ ∂E, p|p−1 |C|, ib(ξ)) represents S(j ∗ x), so j ∗ S(x) = S(j ∗ x). This completes the proof of Lemma 7. Recall that wI ∈ IF (B P LF ), wH ∈ HF (BGF ) are the universal elements. There is a based map χ : B P LF −−→BGF such that S(wI ) = χ∗ (wH ). This defines the based homotopy class of χ uniquely. Consider the topological space L = {(x, ψ)} of pairs with x ∈ B P LF , ψ : I−−→BGF such that χ(x) = ψ(0), ψ(1) = bpt, with (bpt,constant) as base-point.