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Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

ପାଠ୍ୟ

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.

ଅଧିକ ପାଠ୍ୟ

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

ଏହି ଶିକ୍ଷାଗୁଡ଼ିକ ମଧ୍ୟରୁ କିଛି ଖୋଲା ପାଠ୍ୟରୁ ଆୟୋଜିତ ହୋଇଛି: Wikipedia. ପ୍ରତ୍ୟେକ ଶିକ୍ଷାଦାନ ପୃଷ୍ଠା ଏହାର ଉତ୍ସ ଏବଂ ଲାଇସେନ୍ସ ନାମ ଦେଇଥାଏ ।

ଏଠାରେ ବ୍ୟବହୃତ ପ୍ରତୀକଗୁଡିକ

ସମ୍ପୂର୍ଣ୍ଣ ବର୍ଣ୍ଣନା ପାଇଁ ଯେକୌଣସି ପ୍ରତୀକକୁ ଟ୍ୟାପ କରନ୍ତୁ, ଗୋଟିଏ ଚିତ୍ର, ଏବଂ ଏଥିରେ ଥିବା ପ୍ରତ୍ୟେକ ଅକ୍ଷରର ଅର୍ଥ କଣ।

ଲୋକମାନେ ପଚାରୁଥିବା ପ୍ରଶ୍ନ

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).

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