Colloquium publications.

AUTHOR'S PREFACE
The Cambridge Colloquium lectures on Analysis Situs were
intended as an introduction to the problem of discovering the
n-dimensional manifolds and characterizing them by means of
invariants. For the present publication the material of the
lectures has been thoroughly revised and is presented in a more
formal way. It thus constitutes something like a systematic
treatise on the elements of Analysis Situs. The author does not,
however, imagine that it is in any sense a definitive treatment.
For the subject is still in such a state that the best welcome which
can be offered to any comprehensive treatment is to wish it a
speedy obsolescence.
The definition of a manifold which has been used is that which
proceeds from the consideration of a generalized polyhedron
consisting of n-dimensional cells. The relations among the cells
are described by means of matrices of integers and the properties
of the manifolds are obtained by operations with the matrices.
The most important of these matrices were introduced by H.
Poincare to whom we owe most of our knowledge of n-dimensional
manifolds* for the cases in which n > 2. But it is also found
convenient to employ certain more elementary matrices of incidence whose elements are reduced modulo 2, and from which
the Poincare matrices can be derived.
The operations on the matrices lead to combinatorial results
which are independent of the particular way in which a manifold is divided into cells and therefore lead to theorems of
Analysis Situs. The proof that this is so is based on an article
by J. W. Alexander in the Transactions of the American Math* Poincare's work is contained in the following four memoirs: Analysis
Situs, Journal de l'Ecole Polytechnique, 2d Ser., Vol. 1 (1895); Complement
a l'Analysis Situs, Rendiconti del Circolo Matematico di Palermo, Vol. 13
(1899); Second Complement, Proceedings of the London Mathematical
Society, Vol. 32 (1900); Cinquieme Complement, Rendiconti, Vol. 18 (1904).
The third and fourth Complements deal with applications to Algebraic
Geometry, into which we do not go.
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