Let ℝ∞ be the direct limit of the Euclidean spaces ℝn. Now the orthogonal group O(∞) acts on ℝn and the direct limit O(∞) of the groups O(∞) acts on ℝ∞. The infinite pin group Pin(∞) is an extension of ℤ2 by O(∞) and admits the following presentation: the generators are the unit vectors xf in ℝ∞ and the relations are

Let G and H be Hausdorff locally compact groups. By R(G, H) we denote the space of continuous homomorphisms of G into H equipped with the compact-open topology, namely that which is generated by subsets of the form

where K is any compact subset of G and U is any open subset of H. Further, let R0(G, H) be the subset of R(G, H) consisting of elements r satisfying the conditions that the quotient space H/r(G) is compact and r is proper, i.e., the action of G on H defined as the action by left translations of r(G) on H is proper. It has been shown by H. Abels [2] that if G and H are such that at least one of them has a compact defining subset then R0(G,H) is open in R(G,H). Moreover, for each r0 ∈ R0(G,H) there exists a neighbourhood M and a compact subset F of H such that r(G) F = H for all r ∈ M and for each compact subset K of H the union is a relatively compact subset of G. It is furthermore shown in [1] that if G contains no small subgroups and H is connected then the subset of R0(G, H) consisting of isomorphisms of G into H is open in R(G, H). These results, in the case in which H is a connected Lie Group and G is a discrete group, have been established by Weil in [5] and [6] appendix 1.

Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney [11]. Lie groups are parallelizable and so by the theorem of M. W. Hirsch [5] there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger [3]. Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.