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• W2019558487 abstract "A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript normal infinity> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{C^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a base point <italic>p</italic>, we construct an algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1 Baseline equals pi Superscript 1 Baseline left-parenthesis German upper M comma p right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1} = {pi ^1}(mathfrak {M},p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of iterated integrals, whose value along each loop at <italic>p</italic> depends only on the homotopy class of the loop. Thus <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be taken as a commutative algebra of functions on the fundamental group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi 1 left-parenthesis German upper M right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi _1}(mathfrak {M})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose multiplication induces a comultiplication <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1 Baseline right-arrow pi Superscript 1 Baseline circled-times pi Superscript 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1} to {pi ^1} otimes {pi ^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which makes <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a Hopf algebra. The algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Superscript 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>π<!-- π --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{pi ^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra <italic>A</italic> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript normal infinity> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{C^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; (b) the <italic>A</italic>-module <italic>M</italic> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript normal infinity> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{C^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 1-forms on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; (c) the usual differentiation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d colon upper A right-arrow upper M> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d:A to M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and (d) the evaluation map at the base point <italic>p</italic>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon colon upper A right-arrow upper K> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon :A to K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>K</italic> being the real (or complex) number field." @default.
• W2019558487 created "2016-06-24" @default.
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• W2019558487 date "1971-01-01" @default.
• W2019558487 modified "2023-10-18" @default.
• W2019558487 title "Algebras of iterated path integrals and fundamental groups" @default.
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