1. $H_0$ and reduced homology (5/2/2024).

$H_0$ and reduced homology

due by Monday, Feb 5, 2024

Let $X$ be a topological space and $P$ the set of path-components of $X$. Let $\Z[P]$ be the free abelian group generated by $P$. Recall that $\Delta^0$ is a single point and singular $0$-simplicies in $X$ are maps $\sigma: \Delta^0 \to X$, which can thus be identified with points.

Define a homomorphism $\Phi: C_0(X) \to \Z[P]$ by sending a singular $0$-simplex $\sigma$ to the path-component $P_\sigma$ of $X$ containing the image of $\sigma$.

  1. Show that $\Phi$ induces a homomorphism $\varphi: H_0(X) \to \Z[P]$.
  2. Show that $\varphi$ is injective.
  3. Show that $\varphi$ is surjective.

Recall that $\widetilde{C}_0(X) = C_0(X)$, $\widetilde{C}_{-1}(X) = \Z$ and $\widetilde{\partial}_0 = \varepsilon$ is the augmentation map defined by $\varepsilon(\sigma) = 1$ for a singular $0$-simplex $\sigma$.

  1. Show that there is an injective homomorphism $i: \widetilde{H}_0(X) \to H_0(X)$ induced by the identity on $\widetilde{C}_0(X) = C_0(X)$.
  2. Show that $\varepsilon$ induces a homomorphism $\bar{\varepsilon}:H_0(X) \to \Z$.
  3. Show that $\bar{\varepsilon}$ is surjective if and only if $X$ is non-empty.
  4. Show that the image of $i: \widetilde{H}_0(X) \to H_0(X)$ is the kernel of $\bar{\varepsilon}$.
  5. Suppose $X$ is non-empty. Show that there exists a homomorphism $s: \Z \to \widetilde{H}_0(X)$ (called a section) such that $\bar{\varepsilon} \circ s = 1_\Z$.

Define the homomorphism $\psi: \widetilde{H}_0(X) \times \Z \to H_0(X)$ by $\psi(z, n) = i(z) + s(n)$.

  1. Show that $\psi$ is injective.
  2. Show that $\psi$ is surjective.
  3. What is $\widetilde{H}_0(\phi)$? ($\phi$ is the empty space)
  4. What is $\widetilde{H}_{-1}(\phi)$?
  5. What is $\widetilde{H}_{-1}(X)$ if $X$ is non-empty?