In the paper

R. Brown and P.J. Higgins Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41.

we generalised to all dimensions the proof formulation and result proof due to Crowell (1959) of the classical Seifert-van Kampen Theorem for the fundamental group. The Our generalised proof did not involve singular homology, nor simplicial approximation. Consequences were:

1. The Brouwer Degree Theorem (determination for $n \geqslant 1$ of $\pi_r(S^n)$ for $r \leqslant n$)n$; and also the determination of$\pi_n(S^n \vee W)$where$W$is a wedge of circles (without using covering spaces); 2. the Relative Hurewicz Theorem (as a determination of$\pi_n(X \cup CA,x)$if$(X,A)$is$(n-1)$-connected); 3. J.H.C. Whitehead's determination of$\pi_2(X \cup e^2_\lambda,X,x)$as the free crossed$\pi_1(X,x)$-module on the$2$-cells$e^2_\lambda$(Whitehead's proof involved transversality and knot theory, and there are maybe three other proofs); 4. a generalisation of 3. to determine$\pi_2(X \cup cup_f CA, X,x)$as an induced a crossed module induced by$f_*: \pi_1(A,a) \to \pi_1(X,x)$(at for$A,X$connected --at present no other proof is available). The full story is also given in the EMS Tract Vol 15. The proofs very much involve the use of filtered spaces, the algebra of the multiple compositions of cubes, and the relation with the classical invariants of relative homotopy groups. 2 slight wording change In the paper R. Brown and P.J. Higgins Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41. we proved a generalisation generalised to all dimensions the proof and result of the classical Seifert-van Kampen Theorem for the fundamental group. The proof did not involve singular homology, nor simplicial approximation. Consequences were: 1. The Brouwer Degree Theorem (determination of$\pi_r(S^n)$for$r \leqslant n$); 2. the Relative Hurewicz Theorem (as a determination of$\pi_n(X \cup CA,x)$if$(X,A)$is$(n-1)$-connected); 3. J.H.C. Whitehead's determination of$\pi_2(X \cup e^2_\lambda,X,x)$as the free crossed$\pi_1(X,x)$-module on the$2$-cells$e^2_\lambda$; 4. a generalisation of 3. to determine$\pi_2(X \cup CA, X,x)\$ as an induced crossed module (at present no other proof is available).

The full story is also given in the EMS Tract Vol 15.

The proofs very much involve the use of filtered spaces, the algebra of the multiple compositions of cubes, and the relation with the classical invariants of relative homotopy groups.

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