My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion

$$C_\bullet^\mathcal{U}(X)\hookrightarrow C_\bullet(X)$$

induces an isomorphism in singular homology

$$H_p^\mathcal{U}(X)\cong H_p(X)$$

For all $p\geq 0$. In both references, a chain homotopy $h:C_p(X)\longrightarrow C_{p+1}(X)$ between the barycentric subdivision operador and the identity map is given by

If $p=0$, $h$ is the zero homomorphism. If we have defined $h$ up to some $p\in\mathbb{N}$, and $\sigma$ is a $p-$simplex in $X$, then

 

$$h\sigma=\sigma_\#b_p*(i_p-si_p-h\partial i_p)$$

 

Where $b_p$ is the barycentre of the standard $p-$simplex $\Delta_p$ and $*$ is the cone operator. We then extend $h$ linearly to singular chains: $h\big(\sum_{i\in I}n_i\sigma_i\big)=\sum_{i\in I}n_ih\sigma_i$

In contrast with the chain homotopy that appears in the proof of the homotopy axiom, this is really less intuitive, and relies heavily on the equation

$$\partial(w*c)=c-w*\partial c$$

So my questions are:

• How should we understand geometrically this map $h$? What is the geometric intuition that allows us to choose this a a good chain homotopy for our purposes?

• How should we understand the formula $\partial(w*c)=c-w*\partial c$? What is the meaning of this equation geometrically speaking?

• How to come up with such a map in the first place? How has this theorem developed historically?

I understand perfectly both demonstrations, since the calculations are easy to follow; I am just concerned with how this map gives no intuition at first glance about the geometry involved.

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion

$$C_\bullet^\mathcal{U}(X)\hookrightarrow C_\bullet(X)$$

induces an isomorphism in singular homology

$$H_p^\mathcal{U}(X)\cong H_p(X)$$

For all $p\geq 0$. In both references, a chain homotopy $h:C_p(X)\longrightarrow C_{p+1}(X)$ between the barycentric subdivision operador and the identity map is given by

If $p=0$, $h$ is the zero homomorphism. If we have defined $h$ up to some $p\in\mathbb{N}$, and $\sigma$ is a $p-$simplex in $X$, then

$$h\sigma=\sigma_\#b_p*(i_p-si_p-h\partial i_p)$$

Where $b_p$ is the barycentre of the standard $p-$simplex $\Delta_p$ and $*$ is the cone operator. We then extend $h$ linearly to singular chains: $h\big(\sum_{i\in I}n_i\sigma_i\big)=\sum_{i\in I}n_ih\sigma_i$

In contrast with the chain homotopy that appears in the proof of the homotopy axiom, this is really less intuitive, and relies heavily on the equation

$$\partial(w*c)=c-w*\partial c$$

So my questions are:

• How should we understand geometrically this map $h$? What is the geometric intuition that allows us to choose this a a good chain homotopy for our purposes?

• How should we understand the formula $\partial(w*c)=c-w*\partial c$? What is the meaning of this equation geometrically speaking?

• How to come up with such a map in the first place? How has this theorem developed historically?

I understand perfectly both demonstrations, since the calculations are easy to follow; I am just concerned with how this map gives no intuition at first glance about the geometry involved.

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion

$$C_\bullet^\mathcal{U}(X)\hookrightarrow C_\bullet(X)$$

induces an isomorphism in singular homology

$$H_p^\mathcal{U}(X)\cong H_p(X)$$

For all $p\geq 0$. In both references, a chain homotopy $h:C_p(X)\longrightarrow C_{p+1}(X)$ between the barycentric subdivision operador and the identity map is given by

If $p=0$, $h$ is the zero homomorphism. If we have defined $h$ up to some $p\in\mathbb{N}$, and $\sigma$ is a $(p+1)-$simplex in $X$, then

$$h\sigma=\sigma_\#b_p*(i_p-si_p-h\partial i_p)$$

Where $b_p$ is the barycentre of the standard $p-$simplex $\Delta_p$ and $*$ is the cone operator. We then extend $h$ linearly to singular chains: $h\big(\sum_{i\in I}n_i\sigma_i\big)=\sum_{i\in I}n_ih\sigma_i$

In contrast with the chain homotopy that appears in the proof of the homotopy axiom, this is really less intuitive, and relies heavily on the equation

$$\partial(w*c)=c-w*\partial c$$

So my questions are:

• How should we understand geometrically this map $h$? What is the geometric intuition that allows us to choose this a a good chain homotopy for our purposes?

• How should we understand the formula $\partial(w*c)=c-w*\partial c$? What is the meaning of this equation geometrically speaking?

• How to come up with such a map in the first place? How has this theorem developed historically?

I understand perfectly both demonstrations, since the calculations are easy to follow; I am just concerned with how this map gives no intuition at first glance about the geometry involved.

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion

$$C_\bullet^\mathcal{U}(X)\hookrightarrow C_\bullet(X)$$

induces an isomorphism in singular homology

$$H_p^\mathcal{U}(X)\cong H_p(X)$$

For all $p\geq 0$. In both references, a chain homotopy $h:C_p(X)\longrightarrow C_{p+1}(X)$ between the barycentric subdivision operador and the identity map is given by

If $p=0$, $h$ is the zero homomorphism. If we have defined $h$ up to some $p\in\mathbb{N}$, and $\sigma$ is a $p-$simplex in $X$, then

$$h\sigma=\sigma_\#b_p*(i_p-si_p-h\partial i_p)$$

Where $b_p$ is the barycentre of the standard $p-$simplex $\Delta_p$ and $*$ is the cone operator. We then extend $h$ linearly to singular chains: $h\big(\sum_{i\in I}n_i\sigma_i\big)=\sum_{i\in I}n_ih\sigma_i$

In contrast with the chain homotopy that appears in the proof of the homotopy axiom, this is really less intuitive, and relies heavily on the equation

$$\partial(w*c)=c-w*\partial c$$

So my questions are:

• How should we understand geometrically this map $h$? What is the geometric intuition that allows us to choose this a a good chain homotopy for our purposes?

• How should we understand the formula $\partial(w*c)=c-w*\partial c$? What is the meaning of this equation geometrically speaking?

• How to come up with such a map in the first place? How has this theorem developed historically?

I understand perfectly both demonstrations, since the calculations are easy to follow; I am just concerned with how this map gives no intuition at first glance about the geometry involved.