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Zach L.
Zach L.
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If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong H_n(X;A)$$$$\pi_n(A[X],0) \cong \tilde{H}_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?

If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong H_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?

If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong \tilde{H}_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?

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Zach L.
Zach L.
  • 163
  • 6

Homotopy groups of linearization of a space

If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ where $A$ has discrete topology. Evidently this is true: $$\pi_n(A[X],0) \cong H_n(X;A)$$ if $X$ has the homotopy type of a CW-complex. Can anyone point me to a proof of this fact?