Mathematics - Algebraic Topology, Homology, Cohomology

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data-target="#ref1"><div class="pl-25"><a class="link-gray-900 w-100" href="/science/mathematics">Introduction</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref253521"><div class="d-flex align-items-center"><div class="ml-25"></div><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources">Ancient mathematical sources</a></div><div class="ml-40 toc-drawer sub-toc-drawer"></div></li><li data-target="#ref65969"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65969">Mathematics in ancient Mesopotamia</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65970"><a class="w-100 link-gray-900" href="/science/mathematics/Ancient-mathematical-sources#ref65970">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65971"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems">Geometric and algebraic problems</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65972"><a class="w-100 link-gray-900" href="/science/mathematics/Geometric-and-algebraic-problems#ref65972">Mathematical astronomy</a></li></ul></div></li><li data-target="#ref65973"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt">Mathematics in ancient Egypt</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65974"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65974">The numeral system and arithmetic operations</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65975"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-ancient-Egypt#ref65975">Geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65976"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics">Assessment of Egyptian mathematics</a></li></ul></div></li><li data-target="#ref65977"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65977">Greek mathematics</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65978"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65978">The development of pure mathematics</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65979"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65979">The pre-Euclidean period</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65980"><a class="w-100 link-gray-900" href="/science/mathematics/Assessment-of-Egyptian-mathematics#ref65980">The <em>Elements</em></a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65981"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems">The three classical problems</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65982"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65982">Geometry in the 3rd century <span class="text-smallcaps">bce</span></a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65983"><a class="w-100 link-gray-900" href="/science/mathematics/The-three-classical-problems#ref65983">Archimedes</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65984"><a class="w-100 link-gray-900" href="/science/mathematics/Apollonius">Apollonius</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65985"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry">Applied geometry</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65986"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65986">Later trends in geometry and arithmetic</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref65987"><a class="w-100 link-gray-900" href="/science/mathematics/Applied-geometry#ref65987">Greek trigonometry and mensuration</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65988"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory">Number theory</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref65989"><a class="w-100 link-gray-900" href="/science/mathematics/Number-theory#ref65989">Survival and influence of Greek mathematics</a></li></ul></li></ul></div></li><li data-target="#ref65990"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century">Mathematics in the Islamic world (8th–15th century)</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65991"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65991">Origins</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65992"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65992">Mathematics in the 9th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65993"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-Islamic-world-8th-15th-century#ref65993">Mathematics in the 10th century</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65994"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam">Omar Khayyam</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65995"><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65995">Islamic mathematics to the 15th century</a></li></ul></div></li><li data-target="#ref65996"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Omar-Khayyam#ref65996">European mathematics during the Middle Ages and Renaissance</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref65997"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning">The transmission of Greek and Arabic learning</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65998"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65998">The universities</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref65999"><a class="w-100 link-gray-900" href="/science/mathematics/The-transmission-of-Greek-and-Arabic-learning#ref65999">The Renaissance</a></li></ul></div></li><li data-target="#ref66000"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries">Mathematics in the 17th and 18th centuries</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref66001"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66001">The 17th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66002"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66002">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66003"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-17th-and-18th-centuries#ref66003">Numerical calculation</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66004"><a class="w-100 link-gray-900" href="/science/mathematics/Analytic-geometry">Analytic geometry</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66005" class="has-children"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus">The calculus</a><ul class="list-unstyled" data-level="h4"><li data-target="#ref66006"><a class="w-100 link-gray-900" href="/science/mathematics/The-calculus#ref66006">The precalculus period</a></li></ul><ul class="list-unstyled" data-level="h4"><li data-target="#ref66007"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz">Newton and Leibniz</a></li></ul></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66008"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66008">The 18th century</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66009"><a class="w-100 link-gray-900" href="/science/mathematics/Newton-and-Leibniz#ref66009">Institutional background</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66010"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics">Analysis and mechanics</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66011"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66011">History of analysis</a></li></ul></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref66012"><a class="w-100 link-gray-900" href="/science/mathematics/Analysis-and-mechanics#ref66012">Other developments</a><ul class="list-unstyled" data-level="h3"><li data-target="#ref66013"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations">Theory of equations</a></li></ul><ul class="list-unstyled" data-level="h3"><li data-target="#ref66014"><a class="w-100 link-gray-900" href="/science/mathematics/Theory-of-equations#ref66014">Foundations of geometry</a></li></ul></li></ul></div></li><li data-target="#ref335980"><div class="d-flex align-items-center"><button class="h1-link-drawer-button btn btn-xs btn-circle d-flex rounded" type="button" aria-label="Toggle Heading"><em class="material-icons font-18" data-icon="keyboard_arrow_right"></em></button><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century">Mathematics in the 19th century</a></div><div class="ml-40 toc-drawer sub-toc-drawer"><ul class="list-unstyled" data-level="h2"><li data-target="#ref337058"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337058">Projective geometry</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337059"><a class="w-100 link-gray-900" href="/science/mathematics/Mathematics-in-the-19th-century#ref337059">Making the calculus rigorous</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337060"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series">Fourier series</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337061"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337061">Elliptic functions</a></li></ul><ul class="list-unstyled" data-level="h2"><li data-target="#ref337062"><a class="w-100 link-gray-900" href="/science/mathematics/Fourier-series#ref337062">The theory of numbers</a></li></ul><ul 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</div><span class="marker before-article"></span><section data-level="2" id="ref66032">  <span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph">The early 20th century saw the emergence of a number of theories whose power and utility reside in large part in their generality. Typically, they are marked by an attention to the <a href="https://www.britannica.com/topic/set-mathematics-and-logic" class="md-crosslink autoxref " data-show-preview="true">set</a> or <a href="https://www.britannica.com/science/space-physics-and-metaphysics" class="md-crosslink autoxref " data-show-preview="true">space</a> of all examples of a particular kind. (Functional <a href="https://www.britannica.com/science/analysis-mathematics" class="md-crosslink autoxref " data-show-preview="true">analysis</a> is such an endeavour.) One of the most energetic of these general theories was that of algebraic topology. In this subject a variety of ways are developed for replacing a space by a <a href="https://www.britannica.com/science/group-mathematics" class="md-crosslink autoxref " data-show-preview="true">group</a> and a map between spaces by a map between groups. It is like using X-rays: information is lost, but the shadowy image of the original space may turn out to contain, in an accessible form, enough information to solve the question at hand.</p><span class="marker MOD1 mod-inline"></span> <span class="marker PREMOD2 mod-inline"></span><div class="assemblies"><div class="w-100"><figure class="md-assembly m-0 mb-md-0 card card-borderless print-false" data-assembly-id="2220" data-asm-type="image"><div class="md-assembly-wrapper card-media" data-type="image"><a href="https://cdn.britannica.com/59/2359-004-DF642773/Pieces-surface-octagon-curves.jpg" class="gtm-assembly-link position-relative d-flex align-items-center justify-content-center media-overlay-link card-media" data-href="/media/1/369194/2220"><picture><source media="(min-width: 680px)" srcset="https://cdn.britannica.com/59/2359-004-DF642773/Pieces-surface-octagon-curves.jpg"><img src="https://cdn.britannica.com/59/2359-004-DF642773/Pieces-surface-octagon-curves.jpg?w=300" alt="cutting a Riemann surface" data-width="400" data-height="300" loading="eager"></picture><button class="magnifying-glass btn btn-circle position-absolute shadow btn-white top-10 right-10" aria-label="Zoom in"><em class="material-icons link-blue" data-icon="zoom_in"></em></button></a></div><figcaption class="card-body"><div class="md-assembly-caption text-muted font-14 font-serif line-clamp"><span><a class="gtm-assembly-link md-assembly-title font-weight-bold d-inline font-sans-serif mr-5 media-overlay-link" href="https://cdn.britannica.com/59/2359-004-DF642773/Pieces-surface-octagon-curves.jpg" data-href="/media/1/369194/2220">cutting a Riemann surface</a><span>(Left) Pieces of a surface given by <em>f</em>(<em>x</em>, <em>y</em>) = 0; (right) if the surface is cut along the curves, an octagon is obtained.</span><button class="js-more-btn d-none btn btn-unstyled font-12 bg-white js-content" aria-label="Toggle more/less fact data"><span class="link-blue">(more)</span></button></span></div></figcaption></figure></div></div><p class="topic-paragraph">Interest in this kind of research came from various directions. Galois’s theory of equations was an example of what could be achieved by transforming a problem in one branch of mathematics into a problem in another, more abstract branch. Another <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="impetus" href="https://www.merriam-webster.com/dictionary/impetus" data-type="MW">impetus</a> came from Riemann’s theory of complex functions. He had studied <span id="ref536602"></span><a href="https://www.britannica.com/science/algebraic-function" class="md-crosslink ">algebraic functions</a>—that is, loci defined by equations of the form <em>f</em>(<em>x</em>, <em>y</em>) = 0, where <em>f</em> is a polynomial in <em>x</em> whose coefficients are polynomials in <em>y</em>. When <em>x</em> and <em>y</em> are complex variables, the locus can be thought of as a real <span id="ref536603"></span><a href="https://www.britannica.com/science/surface-geometry" class="md-crosslink " data-show-preview="true">surface</a> spread out over the <em>x</em> plane of complex numbers (today called a <span id="ref536604"></span><a href="https://www.britannica.com/science/Riemann-surface" class="md-crosslink ">Riemann surface</a>). To each value of <em>x</em> there correspond a finite number of values of <em>y</em>. Such surfaces are not easy to comprehend, and Riemann had proposed to draw curves along them in such a way that, if the surface was cut open along them, it could be opened out into a polygonal disk. He was able to establish a profound connection between the <a href="https://www.britannica.com/science/minimum" class="md-crosslink autoxref " data-show-preview="true">minimum</a> number of curves needed to do this for a given surface and the number of functions (becoming <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="infinite" href="https://www.merriam-webster.com/dictionary/infinite" data-type="MW">infinite</a> at specified points) that the surface could then support.</p><span class="marker MOD2 mod-inline"></span> <span class="marker PREMOD3 mod-inline"></span><div class="assemblies"><div class="w-100"><figure class="md-assembly m-0 mb-md-0 card card-borderless print-false" data-assembly-id="90071" data-asm-type="image"><div class="md-assembly-wrapper card-media" data-type="image"><a href="https://cdn.britannica.com/16/80216-004-8416E80C/self-intersection-z--axis.jpg" class="gtm-assembly-link position-relative d-flex align-items-center justify-content-center media-overlay-link card-media" data-href="/media/1/369194/90071"><picture><source media="(min-width: 680px)" srcset="https://cdn.britannica.com/16/80216-004-8416E80C/self-intersection-z--axis.jpg"><img src="https://cdn.britannica.com/16/80216-004-8416E80C/self-intersection-z--axis.jpg?w=300" alt="algebraic topology" data-width="495" data-height="250" loading="eager"></picture><button class="magnifying-glass btn btn-circle position-absolute shadow btn-white top-10 right-10" aria-label="Zoom in"><em class="material-icons link-blue" data-icon="zoom_in"></em></button></a></div><figcaption class="card-body"><div class="md-assembly-caption text-muted font-14 font-serif line-clamp"><span><a class="gtm-assembly-link md-assembly-title font-weight-bold d-inline font-sans-serif mr-5 media-overlay-link" href="https://cdn.britannica.com/16/80216-004-8416E80C/self-intersection-z--axis.jpg" data-href="/media/1/369194/90071">algebraic topology</a><span>(Left) <em>f</em>(<em>x</em>, <em>y</em>) = <em>x</em><sup>2</sup>(<em>x</em> + 1) − <em>y</em><sup>2</sup> = 0 intersects itself at (<em>x</em>, <em>y</em>) = (0, 0). (Right) <em>E</em>(<em>x</em>, <em>y</em>, <em>z</em>) = 0 = <em>y</em><sup>2</sup>(<em>y</em> + <em>z</em><sup>2</sup>) − <em>x</em><sup>2</sup> intersects itself along the <em>z</em>-axis, but the origin is a triple self-intersection. </span><button class="js-more-btn d-none btn btn-unstyled font-12 bg-white js-content" aria-label="Toggle more/less fact data"><span class="link-blue">(more)</span></button></span></div></figcaption></figure></div></div><p class="topic-paragraph">The natural problem was to see how far Riemann’s ideas could be applied to the study of spaces of higher <a href="https://www.britannica.com/science/dimension-geometry" class="md-crosslink autoxref " data-show-preview="true">dimension</a>. Here two lines of inquiry developed. One emphasized what could be obtained from looking at the <a href="https://www.britannica.com/science/projective-geometry" class="md-crosslink autoxref " data-show-preview="true">projective geometry</a> involved. This point of view was fruitfully applied by the <span id="ref536605"></span>Italian school of algebraic geometers. It ran into problems, which it was not wholly able to solve, having to do with the singularities a surface can possess. Whereas a locus given by <em>f</em>(<em>x</em>, <em>y</em>) = 0 may intersect itself only at isolated points, a locus given by an <a href="https://www.britannica.com/science/equation" class="md-crosslink autoxref " data-show-preview="true">equation</a> of the form <em>f</em>(<em>x</em>, <em>y</em>, <em>z</em>) = 0 may intersect itself along curves, a problem that caused considerable difficulties. The second approach emphasized what can be learned from the study of <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="integrals" href="https://www.merriam-webster.com/dictionary/integrals" data-type="MW">integrals</a> along paths on the surface. This approach, pursued by <span id="ref536606"></span><a href="https://www.britannica.com/biography/Charles-Emile-Picard" class="md-crosslink " data-show-preview="true">Charles-Émile Picard</a> and by Poincaré, provided a rich generalization of Riemann’s original ideas.</p><span class="marker MOD3 mod-inline"></span> <span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">On this <a href="https://www.britannica.com/science/base-number-systems" class="md-crosslink autoxref " data-show-preview="true">base</a>, <span id="ref908328"></span><a href="https://www.britannica.com/science/Poincare-conjecture" class="md-crosslink " data-show-preview="true">conjectures</a> were made and a general theory produced, first by Poincaré and then by the American engineer-turned-mathematician <span id="ref536607"></span>Solomon Lefschetz, concerning the nature of <span id="ref536608"></span><a href="https://www.britannica.com/science/manifold" class="md-crosslink " data-show-preview="true">manifolds</a> of arbitrary dimension. Roughly speaking, a <a href="https://www.britannica.com/science/manifold" class="md-crosslink autoxref " data-show-preview="true">manifold</a> is the <em>n</em>-dimensional generalization of the idea of a surface; it is a space any small piece of which looks like a piece of <em>n</em>-dimensional space. Such an object is often given by a single <a href="https://www.britannica.com/science/algebraic-equation" class="md-crosslink autoxref " data-show-preview="true">algebraic equation</a> in <em>n</em> + 1 variables. At first the work of Poincaré and of Lefschetz was concerned with how these <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="manifolds" href="https://www.britannica.com/dictionary/manifolds" data-type="EB">manifolds</a> may be decomposed into pieces, counting the number of pieces and decomposing them in their turn. The result was a list of numbers, called <span id="ref536609"></span><a href="https://www.britannica.com/science/Betti-number" class="md-crosslink ">Betti numbers</a> in honour of the Italian mathematician <span id="ref536610"></span><a href="https://www.britannica.com/biography/Enrico-Betti" class="md-crosslink " data-show-preview="true">Enrico Betti</a>, who had taken the first steps of this kind to extend Riemann’s work. It was only in the late 1920s that the German mathematician <span id="ref536611"></span><a href="https://www.britannica.com/biography/Emmy-Noether" class="md-crosslink " data-show-preview="true">Emmy Noether</a> suggested how the Betti numbers might be thought of as measuring the size of certain groups. At her instigation a number of people then produced a theory of these groups, the so-called <span id="ref536612"></span><a href="https://www.britannica.com/science/homology-mathematics" class="md-crosslink " data-show-preview="true">homology</a> and <span id="ref536613"></span>cohomology groups of a space.</p><span class="marker MOD4 mod-inline"></span> <span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">Two objects that can be deformed into one another will have the same homology and cohomology groups. To assess how much information is lost when a space is replaced by its algebraic topological picture, Poincaré asked the crucial <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="converse" href="https://www.britannica.com/dictionary/converse" data-type="EB">converse</a> question “According to what algebraic conditions is it possible to say that a space is topologically equivalent to a sphere?” He showed by an ingenious example that having the same homology is not enough and proposed a more delicate index, which has since grown into the branch of <a href="https://www.britannica.com/science/topology" class="md-crosslink autoxref " data-show-preview="true">topology</a> called <span id="ref536614"></span><a href="https://www.britannica.com/science/homotopy" class="md-crosslink " data-show-preview="true">homotopy</a> theory. Being more delicate, it is both more basic and more difficult. There are usually standard methods for computing homology and cohomology groups, and they are completely known for many spaces. In contrast, there is scarcely an interesting class of spaces for which all the homotopy groups are known. <a href="https://www.britannica.com/science/Poincare-conjecture" class="md-crosslink autoxref " data-show-preview="true">Poincaré’s conjecture</a> that a space with the homotopy of a <a href="https://www.britannica.com/science/sphere" class="md-crosslink autoxref " data-show-preview="true">sphere</a> actually is a sphere was shown to be true in the 1960s in dimensions five and above, and in the 1980s it was shown to be true for four-dimensional spaces. In 2006 <span id="ref908330"></span><a href="https://www.britannica.com/biography/Grigori-Perelman" class="md-crosslink " data-show-preview="true">Grigori Perelman</a> was awarded a <a href="https://www.britannica.com/science/Fields-Medal" class="md-crosslink " data-show-preview="true">Fields Medal</a> for proving Poincaré’s conjecture true in three dimensions, the only dimension in which Poincaré had studied it.</p><div class="module-spacing"> </div><span class="marker MOD5 mod-inline"></span> </section> <section data-level="2" id="ref66033"> <h2 class="h2">Developments in pure mathematics</h2> <span class="marker PREMOD6 mod-inline"></span><p class="topic-paragraph">The interest in <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="axiomatic" href="https://www.merriam-webster.com/dictionary/axiomatic" data-type="MW">axiomatic</a> systems at the turn of the century led to <a href="https://www.britannica.com/topic/axiom" class="md-crosslink autoxref " data-show-preview="true">axiom</a> systems for the known <span id="ref536615"></span><a href="https://www.britannica.com/science/algebraic-structure" class="md-crosslink ">algebraic structures</a>, that for the theory of fields, for example, being developed by the German mathematician <span id="ref536616"></span><a href="https://www.britannica.com/biography/Ernst-Steinitz" class="md-crosslink ">Ernst Steinitz</a> in 1910. The theory of <span id="ref536617"></span><a href="https://www.britannica.com/science/ring-mathematics" class="md-crosslink " data-show-preview="true">rings</a> (structures in which it is possible to add, subtract, and multiply but not necessarily divide) was much harder to formalize. It is important for two reasons: the theory of <span id="ref536618"></span><a href="https://www.britannica.com/science/algebraic-integer" class="md-crosslink ">algebraic integers</a> forms part of it, because algebraic integers naturally form into rings; and (as Kronecker and Hilbert had argued) <span id="ref536619"></span><a href="https://www.britannica.com/science/algebraic-geometry" class="md-crosslink " data-show-preview="true">algebraic geometry</a> forms another part. The rings that arise there are rings of functions definable on the <a href="https://www.britannica.com/science/curve" class="md-crosslink autoxref " data-show-preview="true">curve</a>, surface, or <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="manifold" href="https://www.britannica.com/dictionary/manifold" data-type="EB">manifold</a> or are definable on specific pieces of it.</p><span class="marker MOD6 mod-inline"></span> <span class="marker PREMOD7 mod-inline"></span><p class="topic-paragraph">Problems in <a href="https://www.britannica.com/science/number-theory" class="md-crosslink autoxref " data-show-preview="true">number theory</a> and algebraic geometry are often very difficult, and it was the hope of mathematicians such as Noether, who laboured to produce a formal, axiomatic theory of rings, that, by working at a more rarefied level, the essence of the concrete problems would remain while the distracting special features of any given case would fall away. This would make the formal theory both more general and easier, and to a surprising extent these mathematicians were successful.</p><span class="marker MOD7 mod-inline"></span> <span class="marker PREMOD8 mod-inline"></span><p class="topic-paragraph">A further twist to the development came with the work of the American mathematician <span id="ref536620"></span><a href="https://www.britannica.com/biography/Oscar-Zariski" class="md-crosslink ">Oscar Zariski</a>, who had studied with the Italian school of algebraic geometers but came to feel that their method of working was imprecise. He worked out a detailed program whereby every kind of geometric configuration could be redescribed in algebraic terms. His work succeeded in producing a rigorous theory, although some, notably Lefschetz, felt that the <a href="https://www.britannica.com/science/geometry" class="md-crosslink autoxref " data-show-preview="true">geometry</a> had been lost sight of in the process.</p><span class="marker MOD8 mod-inline"></span> <span class="marker PREMOD9 mod-inline"></span><p class="topic-paragraph">The study of algebraic geometry was <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="amenable" href="https://www.merriam-webster.com/dictionary/amenable" data-type="MW">amenable</a> to the topological methods of Poincaré and Lefschetz so long as the manifolds were defined by equations whose coefficients were <span id="ref536621"></span><a href="https://www.britannica.com/science/complex-number" class="md-crosslink " data-show-preview="true">complex numbers</a>. But, with the creation of an abstract theory of <span id="ref536622"></span><a href="https://www.britannica.com/science/field-mathematics" class="md-crosslink ">fields</a>, it was natural to want a theory of varieties defined by equations with coefficients in an arbitrary field. This was provided for the first time by the French mathematician <span id="ref536623"></span><a href="https://www.britannica.com/biography/Andre-Weil" class="md-crosslink " data-show-preview="true">André Weil</a>, in his <em><span id="ref536624"></span>Foundations of Algebraic Geometry</em> (1946), in a way that drew on Zariski’s work without suppressing the intuitive appeal of geometric concepts. Weil’s theory of <span id="ref536625"></span><a href="https://www.britannica.com/science/polynomial" class="md-crosslink " data-show-preview="true">polynomial</a> equations is the proper setting for any investigation that seeks to determine what properties of a geometric object can be <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="derived" href="https://www.britannica.com/dictionary/derived" data-type="EB">derived</a> solely by algebraic means. But it falls tantalizingly short of one topic of importance: the solution of polynomial equations in integers. This was the topic that Weil took up next.</p><span class="marker MOD9 mod-inline"></span> <span class="marker PREMOD10 mod-inline"></span><p class="topic-paragraph">The central difficulty is that in a field it is possible to divide but in a <a href="https://www.britannica.com/science/ring-mathematics" class="md-crosslink autoxref " data-show-preview="true">ring</a> it is not. The integers form a ring but not a field (dividing 1 by 2 does not yield an <a href="https://www.britannica.com/science/integer" class="md-crosslink autoxref " data-show-preview="true">integer</a>). But Weil showed that simplified versions (posed over a field) of any question about integer solutions to polynomials could be profitably asked. This transferred the questions to the domain of algebraic geometry. To count the number of solutions, Weil proposed that, since the questions were now geometric, they should be amenable to the techniques of algebraic topology. This was an <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="audacious" href="https://www.merriam-webster.com/dictionary/audacious" data-type="MW">audacious</a> move, since there was no suitable theory of algebraic topology available, but Weil conjectured what results it should yield. The difficulty of Weil’s conjectures may be judged by the fact that the last of them was a generalization to this setting of the famous <span id="ref536626"></span><a href="https://www.britannica.com/science/Riemann-hypothesis" class="md-crosslink " data-show-preview="true">Riemann hypothesis</a> about the <a href="https://www.britannica.com/science/Riemann-zeta-function" class="md-crosslink " data-show-preview="true">zeta function</a>, and they rapidly became the focus of international attention.</p><span class="marker MOD10 mod-inline"></span> <span class="marker PREMOD11 mod-inline"></span><p class="topic-paragraph">Weil, along with <span id="ref536627"></span>Claude Chevalley, <span id="ref536628"></span><a href="https://www.britannica.com/biography/Henri-Cartan" class="md-crosslink " data-show-preview="true">Henri Cartan</a>, <span id="ref536629"></span><a href="https://www.britannica.com/biography/Jean-Dieudonne" class="md-crosslink " data-show-preview="true">Jean Dieudonné</a>, and others, created a group of young French mathematicians who began to publish virtually an encyclopaedia of mathematics under the name <span id="ref536630"></span><a href="https://www.britannica.com/topic/Nicolas-Bourbaki" class="md-crosslink " data-show-preview="true">Nicolas Bourbaki</a>, taken by Weil from an obscure general of the <a href="https://www.britannica.com/event/Franco-German-War" class="md-crosslink autoxref " data-show-preview="true">Franco-German War</a>. Bourbaki became a self-selecting group of young mathematicians who were strong on <a href="https://www.britannica.com/science/algebra" class="md-crosslink autoxref " data-show-preview="true">algebra</a>, and the individual Bourbaki members were interested in the Weil conjectures. In the end they succeeded completely. A new kind of algebraic topology was developed, and the Weil conjectures were proved. The generalized Riemann <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="hypothesis" href="https://www.merriam-webster.com/dictionary/hypothesis" data-type="MW">hypothesis</a> was the last to surrender, being established by the Belgian <span id="ref536631"></span><a href="https://www.britannica.com/biography/Pierre-Deligne" class="md-crosslink " data-show-preview="true">Pierre Deligne</a> in the early 1970s. Strangely, its resolution still leaves the original Riemann hypothesis unsolved.</p><span class="marker MOD11 mod-inline"></span> <span class="marker PREMOD12 mod-inline"></span><p class="topic-paragraph">Bourbaki was a key figure in the rethinking of structural mathematics. Algebraic topology was <span id="ref536632"></span>axiomatized by <span id="ref536633"></span><a href="https://www.britannica.com/biography/Samuel-Eilenberg" class="md-crosslink ">Samuel Eilenberg</a>, a Polish-born American mathematician and Bourbaki member, and the American mathematician <span id="ref536634"></span>Norman Steenrod. <span id="ref536635"></span><a href="https://www.britannica.com/biography/Saunders-Mac-Lane" class="md-crosslink " data-show-preview="true">Saunders Mac Lane</a>, also of the <a href="https://www.britannica.com/place/United-States" class="md-crosslink autoxref " data-show-preview="true">United States</a>, and Eilenberg extended this axiomatic approach until many types of mathematical structures were presented in families, called categories. Hence there was a <span id="ref536636"></span><a href="https://www.britannica.com/science/category-mathematics" class="md-crosslink ">category</a> consisting of all groups and all maps between them that preserve multiplication, and there was another category of all topological spaces and all <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="continuous" href="https://www.britannica.com/dictionary/continuous" data-type="EB">continuous</a> maps between them. To do algebraic topology was to transfer a problem posed in one category (that of topological spaces) to another (usually that of commutative groups or rings). When he created the right algebraic topology for the Weil conjectures, the German-born French mathematician <span id="ref536637"></span><a href="https://www.britannica.com/biography/Alexandre-Grothendieck" class="md-crosslink " data-show-preview="true">Alexandre Grothendieck</a>, a Bourbaki of enormous energy, produced a new description of algebraic geometry. In his hands it became infused with the language of category theory. The route to algebraic geometry became the steepest ever, but the views from the summit have a naturalness and a profundity that have brought many experts to prefer it to the earlier formulations, including Weil’s.</p><span class="marker MOD12 mod-inline"></span> <span class="marker PREMOD13 mod-inline"></span><p class="topic-paragraph">Grothendieck’s formulation makes algebraic geometry the study of equations defined over rings rather than fields. Accordingly, it raises the possibility that questions about the integers can be answered directly. Building on the work of like-minded mathematicians in the United States, France, and Russia, the German <span id="ref536638"></span><a href="https://www.britannica.com/biography/Gerd-Faltings" class="md-crosslink " data-show-preview="true">Gerd Faltings</a> triumphantly <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="vindicated" href="https://www.merriam-webster.com/dictionary/vindicated" data-type="MW">vindicated</a> this approach when he solved the Englishman Louis <span id="ref536639"></span><a href="https://www.britannica.com/science/Mordells-conjecture" class="md-crosslink ">Mordell’s conjecture</a> in 1983. This conjecture states that almost all polynomial equations that define curves have at most finitely many rational solutions; the cases excluded from the conjecture are the simple ones that are much better understood.</p><span class="marker MOD13 mod-inline"></span> <span class="marker PREMOD14 mod-inline"></span><p class="topic-paragraph">Meanwhile, <span id="ref536640"></span>Gerhard Frey of Germany had pointed out that, if <a href="https://www.britannica.com/science/Fermats-last-theorem" class="md-crosslink " data-show-preview="true">Fermat’s last theorem</a> is false, so that there are integers <em>u</em>, <em>v</em>, <em>w</em> such that <em>u</em><sup><em>p</em></sup> + <em>v</em><sup><em>p</em></sup> = <em>w</em><sup><em>p</em></sup> (<em>p</em> greater than 5), then for these values of <em>u</em>, <em>v</em>, and <em>p</em> the curve <em>y</em><sup>2</sup> = <em>x</em>(<em>x</em> − <em>u</em><sup><em>p</em></sup>)(<em>x</em> + <em>v</em><sup><em>p</em></sup>) has properties that contradict major <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="conjectures" href="https://www.britannica.com/dictionary/conjectures" data-type="EB">conjectures</a> of the Japanese mathematicians Taniyama Yutaka and Shimura Goro about elliptic curves. Frey’s observation, refined by <a href="https://www.britannica.com/biography/Jean-Pierre-Serre" class="md-crosslink " data-show-preview="true">Jean-Pierre Serre</a> of France and proved by the American Ken Ribet, meant that by 1990 Taniyama’s unproven conjectures were known to imply <span id="ref536641"></span><a href="https://www.britannica.com/science/Fermats-last-theorem" class="md-crosslink " data-show-preview="true">Fermat’s last theorem</a>.</p><span class="marker MOD14 mod-inline"></span> <span class="marker PREMOD15 mod-inline"></span><p class="topic-paragraph">In 1993 the English mathematician <span id="ref536642"></span><a href="https://www.britannica.com/biography/Andrew-Wiles" class="md-crosslink " data-show-preview="true">Andrew Wiles</a> established the <span id="ref536643"></span>Shimura-Taniyama conjectures in a large range of cases that included Frey’s curve and therefore Fermat’s last theorem—a major feat even without the connection to Fermat. It soon became clear that the argument had a serious flaw; but in May 1995 Wiles, assisted by another English mathematician, <a href="https://www.britannica.com/biography/Richard-Taylor-filmmaker" class="md-crosslink autoxref " data-show-preview="true">Richard Taylor</a>, published a different and valid approach. In so doing, Wiles not only solved the most famous outstanding <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="conjecture" href="https://www.britannica.com/dictionary/conjecture" data-type="EB">conjecture</a> in mathematics but also triumphantly vindicated the sophisticated and difficult methods of modern number theory.</p><span class="marker MOD15 mod-inline"></span> </section> <span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":25,"pagesTotal":27,"pageId":369194,"pageLength":2086,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/libs/jquery-3.5.0.min.js?v=3.133.9"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.js?v=3.133.9' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"D","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"PAGINATED","gisted":false,"pageNumber":25,"hasSummarizeButton":false,"hasAskButton":true} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

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