By von Oniscik A.L., Sulanke R.

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**Example text**

73–84]) construct a regular homotopy of the f i which leaves fixed all except a neighbourhood of the arc α, and ‘pulls’ this across δ to the ‘other side’ of α′ , thus getting rid of the intersection point P and introducing no new undesirable points. Proceeding in this manner, we convert the f i (and the fi ) into disjoint embeddings. Now perform handle subtraction : let N0 be obtained from N by deleting the interiors of the fi (Dk × Dk ): let U be the union of the images of the fi : let M0 be obtained from M .

However, his methods are of no use in relativising our result : the functors Lm do not appear to be effaceable. 3. statement of results 37 We now state the geometrical part of our results. 2. Let X be a simple Poincar´e (n + 1)-ad, M a compact manifold (n + 1)-ad, φ : M → X a map of degree 1; ν a bundle over |X|, F a stable trivialisation of τ|M| ⊕ φ∗ ν. Then there is an obstruction θ in Ldim |M| π(X) which depends only on the bordism class of (M, φ, F ) and vanishes if that class contains a simple homotopy equivalence; conversely, if dim M { } > 4, its vanishing is sufficient.

Then the diagram H r (M ; φ∗ B) o φ∗ [M ] ∩ t Hm−r (M ; φ∗ B) φ∗ H r (X; B) [X] ∩ t G Hm−r (X; B) is commutative, so [M ] ∩ induces an isomorphism of the cokernel K r (M ; φ∗ B) t of φ∗ on the kernel Km−r (M ; φ∗ B) of φ∗ . Thus if φ is k-connected, φ∗ and ∗ φ are isomorphisms for r < k and for r > m − k. Similarly let φ : (N, M ) → (Y, X) be a map of degree 1 of Poincar´e pairs. Then φ∗ gives split surjections of homology groups M → X, N → Y and (N, M ) → (Y, X) with kernels K∗ , and split injections of cohomology with cokernels K ∗ .