{"product_id":"against-the-grain-book-boris-nikolayevich-yeltsin-9780671700553","title":"Against the Grain","description":"\u003cp\u003eThis book contains an in-depth overview of the current state of the recently emerged and rapidly growing theory of \u003cem\u003eG\u003csub\u003en\u003c\/sub\u003e\u003csup\u003ek\u003c\/sup\u003e\u003c\/em\u003e groups, picture-valued invariants, and braids for arbitrary manifolds. Equivalence relations arising in low-dimensional topology and combinatorial group theory inevitably lead to the study of invariants, and good invariants should be strong and apparent. An interesting case of such invariants is picture-valued invariants, whose values are not algebraic objects, but geometrical constructions, like graphs or polyhedra.\u003c\/p\u003e\u003cp\u003eIn 2015, V O Manturov defined a two-parametric family of groups \u003cem\u003eG\u003csub\u003en\u003c\/sub\u003e\u003csup\u003ek\u003c\/sup\u003e\u003c\/em\u003e and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in \u003cem\u003eG\u003csub\u003en\u003c\/sub\u003e\u003csup\u003ek\u003c\/sup\u003e\u003c\/em\u003e.\u003c\/p\u003e\u003cp\u003eThe book is devoted to various realisations and generalisations of this principle in the broad sense. The groups \u003cem\u003eG\u003csub\u003en\u003c\/sub\u003e\u003csup\u003ek\u003c\/sup\u003e\u003c\/em\u003e have many epimorphisms onto free products of cyclic groups; hence, invariants constructed from them are powerful enough and easy to compare. However, this construction does not work when we try to deal with points on a 2-surface, since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups -- \u003cem\u003eΓ\u003csub\u003en\u003c\/sub\u003e\u003csup\u003ek\u003c\/sup\u003e\u003c\/em\u003e, which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds.\u003c\/p\u003e","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":49525444509969,"sku":"CIN0671700553G","price":6.08,"currency_code":"USD","in_stock":false},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":49531718205713,"sku":"GOR001424042","price":19.49,"currency_code":"USD","in_stock":true},{"title":"GB \/ WELL_READ \/ INTERNAL","offer_id":49895089504529,"sku":"GOR008597773","price":0.0,"currency_code":"USD","in_stock":false},{"title":"US \/ WELL_READ \/ SBYB","offer_id":50256087777553,"sku":"CIN0671700553A","price":4.69,"currency_code":"USD","in_stock":false},{"title":"US \/ VERY_GOOD \/ SBYB","offer_id":50762545561873,"sku":"CIN0671700553VG","price":6.32,"currency_code":"USD","in_stock":false},{"title":"GB \/ GOOD \/ INTERNAL","offer_id":50986541678865,"sku":"GOR007646169","price":10.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0671700553.jpg?v=1750978102","url":"https:\/\/www.worldofbooks.com\/products\/against-the-grain-book-boris-nikolayevich-yeltsin-9780671700553","provider":"World of Books ","version":"1.0","type":"link"}