"Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist’s field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems." (Samuel Eilenberg &Norman E Steenrod, "Foundations of Algebraic Topology", 1952)
"The philosophical emphasis here is: to solve a geometrical
problem of a global nature, one first reduces it to a homotopy theory problem;
this is in turn reduced to an algebraic problem and is solved as such. This path
has historically been the most fruitful one in algebraic topology. (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)
"The various homology and cohomology theories appear as complicated
machines, the end product of which is an assignment of a graded group to a
topological space, through a series of processes which look so arbitrary that
one wonders why they succeed at all." (Jean Dieudonné, "A History of Algebraic
and Differential Topology, 1900 - 1960", 1989)
"A homeomorphism may be thought of as the best possible type of continuous function, and homeomorphic spaces are considered the same in topology. [...] A complex has two structures: that of the topological space underlying the complex, and the subdivision of the complex into cells. The corresponding functions will have to preserve this dual nature. In particular, it would be nice if these functions, yet to be defined, would also induce nice functions on the homology of the complex. Since homology has an algebraic group structure, we want the functions to induce homomorphisms on the homology groups. The best of all functions would be homeomorphisms on the underlying topological space and also isomorphisms on the homology groups." (L Christine Kinsey. "Topology of Surfaces", 1993)
"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 I Monastyrsky, "Topology of Gauge Fields and Condensed Matter", 1993)
"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 greatest successes of the combinatorial topology has been the extension of Homology Theory to general topological spaces. In what discussed above it is clear that Homology groups can be defined for a special kind of space, namely, compact polyhedron and the complexes obtained there were finite althrough. Singular Homology theory extends the notion of Homology groups for general topological spaces by associating with each space a chain complex. A continuous map induces homology homomorphisms in an obvious way and as a consequence it follows that homotopic maps induce the same homomorphisms. There is a natural homomorphism also from homology groups to singular homology groups. In the following lines we give a sketch of notions relevant in singular homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)
"The exquisite world of algebraic topology came into existence out of our attempts to solve topological problems by the use of algebraic tools and this revealed to us the nice interplay between algebra and topology which causes each to reinforce interpretations of the other there by breaking down the often artificial subdivision of mathematics into different branches and emphasizing the essential unity of all mathematics. The homology theory is the main branch of algebraic topology and plays the main role in the classification problems of topological spaces. There are various approaches to the study of this theory such as geometric approach, abstract approach and axiomatic approach. Since geometric approach appeals easily to our intuition, we shall start with geometric approach, the relevant development being called simplicial homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)
"Although it is not difficult to count the holes in a real pretzel in your hand, prior to eating it, when a surface pops out of an abstract mathematical construction it can be very difficult to figure out its properties, such as how many holes it has. The cohomology groups can help us to do so." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"Topology is a child of twentieth century mathematical thinking. It allows us to consider the shape and structure of an object without being wedded to its size or to the distances between its component parts. Knot theory, homotopy theory, homology theory, and shape theory are all part of basic topology. It is often quipped that a topologist does not know the difference between his coffee cup and his donut - because each has the same abstract 'shape' without looking at all alike." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
"Homology and cohomology don’t tell us everything we would like to know about the shape of a topological space - distinct spaces can have the same homology and cohomology – but they do provide a lot of useful information, and a systematic framework in which to calculate it and use it." (Ian Stewart, "Visions of Infinity", 2013)
"In homotopy, we ask whether a given loop can be shrunk continuously to a point. In homology, we ask a different question: does the loop form the boundary of a topological disc? That is, can you fit one or more triangular patches together so that the result is a region without any holes, and the boundary of this region is the loop concerned?" (Ian Stewart, "Visions of Infinity", 2013)
"At first, topology can seem like an unusually imprecise branch of mathematics. It’s the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. […] While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds. […] homology infers an object’s holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)
"Homology translates this world of vague shapes into the rigorous world of algebra, a branch of mathematics that studies particular numerical structures and symmetries. Mathematicians study the properties of these algebraic structures in a field known as homological algebra. From the algebra they indirectly learn information about the original topological shape of the data. Homology comes in many varieties, all of which connect with algebra." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)
"Mathematicians extract a shape’s homology from its chain complex, which provides structured data about the shape’s component parts and their boundaries - exactly what you need to describe holes in every dimension. […] The definition of homology is rigid enough that a computer can use it to find and count holes, which helps establish the rigor typically required in mathematics. It also allows researchers to use homology for an increasingly popular pursuit: analyzing data." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)
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