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Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

පාඩම්

Advanced The fundamental group Loops up to deformation; the circle's winding number; van Kampen. polar form of -1 Advanced Homology and the Euler characteristic Counting holes in every dimension with chain complexes. 6 - 12 + 8

Attach a group to a space so that continuous maps become homomorphisms. Then the question "are these two spaces the same shape?" becomes an algebra computation, which is how we know a doughnut is not a sphere.

තවත් පාඩම්

Homotopy

Algebraic topology Homotopy Covering space Seifert–Van Kampen theorem Homotopy group Simply connected space Deformation retract

Homology

Homology (mathematics)Simplicial complex Singular homology Chain complex Exact sequence Betti number Euler characteristic CW complex Mayer–Vietoris sequence Cohomology Poincaré duality

Consequences

Brouwer fixed-point theorem Borsuk–Ulam theorem Hairy ball theorem Jordan curve theorem Poincaré conjecture Classification of manifolds

මෙම පාඩම් සමහර විවෘත පෙළ සිට අනුගත කර ඇත: Wikipedia. සෑම පාඩමකම පිටුව එහි මූලාශ්‍රය සහ බලපත්‍රය නම් කරයි.

මෙහිදී භාවිත කරන සංකේත

සම්පූර්ණ අර්ථ දැක්වීම සඳහා ඕනෑම සංකේතයක් ටැප්, පින්තූරයක්, සහ එය සෑම අකුරු අදහස් කරන්නේ කුමක්ද.

ජනතාව අහනවා ප්රශ්න

What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

ඔයාට ඔයාගේම ප්‍රශ්නයක් තියෙනවද?
එය ටයිප් හෝ විසඳුම පිටුව මත එය ඇඳීමට: සෑම පියවරක්, පින්තූරයක්, ඔබ ඉල්ලා තෙක් පිළිතුර සැඟවුණු. විසඳන්න →

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