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Gregory Arone
Gregory Arone
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One approach is to replace the free abelian group functor with the free commutative monoid functor, also known as the infinite symmetric product.

Let $X$ be a pointed CW complex, and let $\mbox{SP}^\infty(X)$ be the infinite symmetric product. By Dold-Thom theorem, if $X$ is connected then the homotopy groups of $\mbox{SP}^\infty(X)$ are the homology groups of $X$.

The advantage of $\mbox{SP}^\infty$ over the free abelian group functor is that it has a natural filtration, and a cell structure that can be analyzed. Given Dold-Thom theorem, the Huriewicz theorem is equivalent to saying that if $X$ is $k-1$-connected (for simplicity let's assume $k>1$), then the map $X\to \mbox{SP}^\infty(X)$ is $k+1$ connected (for simplicity let's assume $k>1$). This can be proved by analysing the cell structure on $\mbox{SP}^\infty(X)$. You can find this in chapter 6 of the book of Aguilar, Gitler and Prieto.

One approach is to replace the free abelian group functor with the free commutative monoid functor, also known as the infinite symmetric product.

Let $X$ be a pointed CW complex, and let $\mbox{SP}^\infty(X)$ be the infinite symmetric product. By Dold-Thom theorem, if $X$ is connected then the homotopy groups of $\mbox{SP}^\infty(X)$ are the homology groups of $X$.

The advantage of $\mbox{SP}^\infty$ over the free abelian functor is that it has a natural filtration, and a cell structure that can be analyzed. Given Dold-Thom theorem, the Huriewicz theorem is equivalent to saying that if $X$ is $k-1$-connected (for simplicity let's assume $k>1$), then the map $X\to \mbox{SP}^\infty(X)$ is $k+1$ connected. This can be proved by analysing the cell structure on $\mbox{SP}^\infty(X)$. You can find this in chapter 6 of the book of Aguilar, Gitler and Prieto.

One approach is to replace the free abelian group functor with the free commutative monoid functor, also known as the infinite symmetric product.

Let $X$ be a pointed CW complex, and let $\mbox{SP}^\infty(X)$ be the infinite symmetric product. By Dold-Thom theorem, if $X$ is connected then the homotopy groups of $\mbox{SP}^\infty(X)$ are the homology groups of $X$.

The advantage of $\mbox{SP}^\infty$ over the free abelian group functor is that it has a natural filtration, and a cell structure that can be analyzed. Given Dold-Thom theorem, the Huriewicz theorem is equivalent to saying that if $X$ is $k-1$-connected, then the map $X\to \mbox{SP}^\infty(X)$ is $k+1$ connected (for simplicity let's assume $k>1$). This can be proved by analysing the cell structure on $\mbox{SP}^\infty(X)$. You can find this in chapter 6 of the book of Aguilar, Gitler and Prieto.

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Gregory Arone
Gregory Arone
  • 10.9k
  • 1
  • 47
  • 66

One approach is to replace the free abelian group functor with the free commutative monoid functor, also known as the infinite symmetric product.

Let $X$ be a pointed CW complex, and let $\mbox{SP}^\infty(X)$ be the infinite symmetric product. By Dold-Thom theorem, if $X$ is connected then the homotopy groups of $\mbox{SP}^\infty(X)$ are the homology groups of $X$.

The advantage of $\mbox{SP}^\infty$ over the free abelian functor is that it has a natural filtration, and a cell structure that can be analyzed. Given Dold-Thom theorem, the Huriewicz theorem is equivalent to saying that if $X$ is $k-1$-connected (for simplicity let's assume $k>1$), then the map $X\to \mbox{SP}^\infty(X)$ is $k+1$ connected. This can be proved by analysing the cell structure on $\mbox{SP}^\infty(X)$. You can find this in chapter 6 of the book of Aguilar, Gitler and Prieto.