SECTIONAL TRANSACTIONS.— A*, B. 399 



invariants of a complex. This completes not only the combinatorial, but 

 also the topological classification of lens spaces. 



Dr. S. EiLENBERG. — On continuous mapping into spheres (11. 15). 



Let M" be a finite or infinite simplicial, orientable, n-dimensional mani- 

 fold ; XcM" a closed and compact sub-set of M» and P*cM» — X a closed 

 (finite or infinite) A-dimensional sub-polyhedron of M". For each A-dimen- 

 sional simplex a* of P*, let s"~*"' he. an {n — k — i)-dimensional spherical 

 manifold, contained in M" — X — P^ and ' simply linked ' with a*. 



Given a continuous mapping /, of M" — P* into an wz-dimensional 

 spherical manifold S'", the mapping /(s"~*~^)cS"' determines a unique 

 element oci(f) of the {n — k — i)th homotopy group 7t„_ft_i(5"'), of S'». 

 We write 



Y*(/) = i:«i(/)«*, 



summed for all the ^-dimensional simplexes of P*. 



(I) Y*(/) is a A-dimensional (finite or infinite) cycle in P*, with coeflfi- 

 cients from the group tzn-k-i (S'"). 

 (II) If y*(/) is homologous to zero in M"— X, there exists a {k — i)- 

 dimensional closed sub-polyhedron P*-^cM" — X, of M», and a 

 continuous mapping f*(M" — P*-i)c5"' such that f{x) =/*(«) 

 for each xsX. 



An application : consider in S" two disjunct sets S^ and 5"""'"', 

 homeomorphic with S^ and 5«-'»-i respectively. S^ is called a retract 

 of S» - 5^-'«-i if there exists a continuous mapping /(5» - 5^~"'"')c5f 

 such that/(x) = x for each xsS^. 



(Ill) 57 is a retract of 5» - S^-""-' if, and only if, the linking coeffi- 

 cient of S^ and 3"^"*'^ is i i (according to the orientations). 



Prof. M. Frechet. — Hilbert space (11.45). 



SECTION B.— CHEMISTRY. 



Thursday, August 18. 



Introduction by Prof. Sir William J. Pope, K.B.E., F.R.S. (lo.o). 



Presidential Address by Prof. C. S. Gibson, O.B.E., F.R.S. , on Recent 

 advances in the chemistry of gold. (See p. 35.) 



Discussion on Recent advances in the organic chemistry of the metals, with 

 special reference to the noble metals. {Exhibition) (11. 15). 



Dr. F. G. Mann. — Introduction. 



Prof. L. O. Brockway (12.0), 



