The functor of singular chains detects weak homotopy equivalences
The functor of singular chains detects weak homotopy equivalences
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奇异链函子检测弱同伦等价
DOI：
10.1090/proc/14555
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发表时间：
2018
影响因子：
1
通讯作者：
M. Zeinalian
中科院分区：
文献类型：
--
作者：
M. Rivera;Felix Wierstra;M. Zeinalian
道路连通的带基点空间\(X\)的正规化奇异链可被视为一个连通的\(E_{\infty}\)-余代数\(\mathbf{C}_*(X)\)，其具有这样的性质：其余棒构造的\(0\)阶同调（它自然是一个余交换双代数）有一个对极；即它是一个余交换霍普夫代数。我们证明，道路连通的带基点空间之间的连续映射\(f:X\rightarrow Y\)是弱同伦等价当且仅当\(\mathbf{C}_*(f):\mathbf{C}_*(X)\rightarrow\mathbf{C}_*(Y)\)是一个\(\mathbf{\Omega}\)-拟同构，即对底层的\(dg\)余结合余代数应用余棒函子\(\mathbf{\Omega}\)之后的\(dg\)代数的拟同构。证明是基于将怀特黑德的一个经典定理与这样的观察相结合：基本群函子以及空间上的局部系统的数据可以从奇异链的代数结构函子性地描述出来。
<p>The normalized singular chains of a path connected pointed space <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
<mml:semantics>
<mml:mi>X</mml:mi>
<mml:annotation encoding="application/x-tex">X</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> may be considered as a connected <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript normal infinity">
<mml:semantics>
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi>
</mml:mrow>
</mml:msub>
<mml:annotation encoding="application/x-tex">E_{\infty }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-coalgebra <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C Subscript asterisk Baseline left-parenthesis upper X right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {C}_*(X)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> with the property that the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0">
<mml:semantics>
<mml:mn>0</mml:mn>
<mml:annotation encoding="application/x-tex">0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>th homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode; i.e., it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper X right-arrow upper Y">
<mml:semantics>
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mo>:</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:mi>Y</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">f: X\to Y</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is a weak homotopy equivalence if and only if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C Subscript asterisk Baseline left-parenthesis f right-parenthesis colon bold upper C Subscript asterisk Baseline left-parenthesis upper X right-parenthesis right-arrow bold upper C Subscript asterisk Baseline left-parenthesis upper Y right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>:</mml:mo>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>Y</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {C}_*(f): \mathbf {C}_*(X)\to \mathbf {C}_*(Y)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is an <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Omega">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">Ω<!-- Ω --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {\Omega }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-quasi-isomorphism, i.e., a quasi-isomorphism of dg algebras after applying the cobar functor <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Omega">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">Ω<!-- Ω --></mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {\Omega }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.</p>