"Homology theory introduces a new connection between
invariants of manifolds. Continuing the 'physical' analogy, we say
that a homology theory studies the intrinsic structure of a manifold by
breaking it into a system of portions arranged simply, or, more precisely, in a
standard way. Then, given certain rules for glueing the portions together, the
theory obtains the whole manifold. The main problem consists in proving the
resultant geometric quantities that are independent of the decomposition and
glueing (i.e., proving the topological invariance of the characteristics)." (Michael
IMonastyrsky, "Topology of Gauge Fields and Condensed Matter", 1993)
"A mathematical idea is fruitful if it makes progress
possible on complex concrete problems left by preceding generations."
"Algebraic topology studies properties of a narrower class of
spaces, - basically the classical objects of mathematics: spaces given by systems
of algebraic and functional equations, surfaces lying in Euclidean space, and
other sets which in mathematics are called manifolds. Examining the narrower
class of spaces permits deeper penetration into their structure."
"But as often happens with rigorous theorems in physics, the
more serious the conclusions which follow from proven assertions, the more
carefully one must examine the initial premises."
"Even the most elegant and beautiful physical theory may disappear without a trace if not confirmed by experiment, while, as a rule, a theorem, once proved, remains in mathematics forever." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"Homology theory studies properties of manifolds by
decomposing them into simpler parts. The structure of these parts can be
investigated easily by introducing algebraic characteristics associated with
these decompositions. The main difficulty lies in proving that the
corresponding characteristics of the decomposition, in fact, do not depend on
the particular choice of the decomposition but are rather a topological invariant
of the manifold itself."
"One of the basic tasks of topology is to learn to
distinguish nonhomeomorphic figures. To this end one introduces the class of
invariant quantities that do not change under homeomorphic transformations of a
given figure. The study of the invariance of topological spaces is connected
with the solution of a whole series of complex questions: Can one describe a
class of invariants of a given manifold? Is there a set of integral invariants
that fully characterizes the topological type of a manifold? and so forth."
"The connection of topology with physics is no passing interlude but rather represents a length affair." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"Topology studies those characteristics of figures which are
preserved under a certain class of continuous transformations. Imagine two
figures, a square and a circular disk, made of rubber. Deformations can convert
the square into the disk, but without tearing the figure it is impossible to
convert the disk by any deformation into an annulus. In topology, this
intuitively obvious distinction is formalized."
"Two figures which can be transformed into one other by
continuous deformations without cutting and pasting are called homeomorphic. […]
The definition of a homeomorphism includes two conditions: continuous and one- to-one
correspondence between the points of two figures. The relation between the two
properties has fundamental significance for defining such a paramount concept
as the dimension of space."
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