Appendix: Unsorted Stuff

  • Assorted info about other Lie Groups:

  • \(O_n, U_n, SO_n, SU_n, Sp_n\)

  • \(\pi_k(U_n) = {\mathbf{Z}}\cdot\indic{k~\text{odd}}\)

    • \(\pi_1(U_n) = 1\)

  • \(\pi_k(SU_n) = {\mathbf{Z}}\cdot\indic{k~\text{odd}}\)

    • \(\pi_1(SU_n) = 0\)

  • \(\pi_k(U_n) = {\mathbf{Z}}/2{\mathbf{Z}}\cdot\indic{k = 0,1\operatorname{mod}8} + {\mathbf{Z}}\cdot\indic{k = 3,7 \operatorname{mod}8}\)

  • \(\pi_k(SP_n) = {\mathbf{Z}}/2{\mathbf{Z}}\cdot\indic{k = 4,5\operatorname{mod}8} + {\mathbf{Z}}\cdot\indic{k = 3,7 \operatorname{mod}8}\)

  • Groups and Group Actions

    • \(\pi_0(G) = G\) for \(G\) a discrete topological group.
    • \(\pi_k(G/H) = \pi_k(G)\) if \(\pi_k(H) = \pi_{k-1}(H) = 0\).
    • \(\pi_1(X/G) = \pi_0(G)\) when \(G\) acts freely/transitively on \(X\).

Cap and Cup Products

\begin{align*} \cup: H^p \times H^q \to H^{p+q}; (a^p \cup b^q)(\sigma) = a^p(\sigma \circ F_p) b^q(\sigma \circ B_q) \end{align*} where \(F_p, B_q\) is embedding into a \(p+q\) simplex.

For \(f\) continuous, \(f^*(a\cup b) = f^*a \cup f^*b\)

It satisfies the Leibniz rule \begin{align*}{\partial}(a^p \cup b^q) = {\partial}a^p \cup b^q + (-1)^p(a^p\cup {\partial}b^q)\end{align*}

\begin{align*} \cap: H_p \times H^q \to H_{p-q}; \sigma \cap \psi = \psi(F\circ\sigma)(B\circ \sigma) \end{align*} where \(F,B\) are the front/back face maps.

Given \(\psi \in C^q, \phi \in C^p, \sigma: \Delta^{p+q} \to X\), we have \begin{align*} \psi(\sigma \cap \phi) = (\phi \cup \psi)(\sigma)\\ {\left\langle {\phi\cup \psi},~{\sigma} \right\rangle} = {\left\langle {\psi},~{\sigma \cap \phi} \right\rangle} \end{align*}

Let \(M^n\) be a closed oriented smooth manifold, and \(A^{\widehat{i}}, B^{\widehat{j}} \subseteq X\) be submanifolds of codimension \(i\) and \(j\) respectively that intersect transversely (so \(\forall p\in A\cap B\), the inclusion-induced map \(T_pA \times T_p B \to T_p X\) is surjective.)

Then \(A\cap B\) is a submanifold of codimension \(i+j\) and there is a short exact sequence \begin{align*} 0 \to T_p(A\cap B) \to T_p A \times T_p B \to T_p X \to 0 \end{align*}

which determines an orientation on \(A\cap B\).

Then the images under inclusion define homology classes

  • \([A] \in H_{\widehat{i}}X\)
  • \([B] \in H_{\widehat{j}}X\)
  • \([A\cap B] \in H_{\widehat{i+j}}X\).

Denoting their Poincare duals by

  • \([A] {}^{ \vee }\in H^i X\)
  • \([B] {}^{ \vee }\in H^j X\)
  • \([A\cap B] {}^{ \vee }\in H^{i+j}X\)

We then have \begin{align*} [A] {}^{ \vee }\smile [B] {}^{ \vee }= [A\cap B] {}^{ \vee }\in H^{i+j} X \end{align*}

Example: in \({\mathbf{CP}}^n\), each even-dimensional cohomology \(H^{2i}{\mathbf{CP}}^n\) has a generator \(\alpha_i\) with is Poincare dual to an \(\widehat{i}\) plane. A generic \(\widehat{i}\) plane intersects a \(\widehat{j}\) plane in a \(\widehat{i+j}\) plane, yielding \(\alpha_i \smile \alpha_j = \alpha_{i+j}\) for \(i+j \leq n\).

Example: For \(T^2\), we have

  • \(H_1T^2 = {\mathbf{Z}}^2\) generated by \([A], [B]\), the longitudinal and meridian circles.
  • \(H_0T^2 = {\mathbf{Z}}\) generated by \([p]\), the class of a point.

Then \(A\cap B = \pm [p]\), and so \begin{align*} [A] {}^{ \vee }\smile [B] {}^{ \vee }= [p] {}^{ \vee }\\ [B] {}^{ \vee }\smile [A] {}^{ \vee }= -[p] {}^{ \vee } \end{align*}

The Long Exact Sequence of a Pair

LES of pair \((A,B) \implies \cdots H_n(B) \to H_n(A) \to H_n(A,B) \to H_{n-1}(B) \cdots\) 

\begin{align*}
\begin{matrix}
&& B & \\
&\diagup &  & \diagdown \\
(A,B) & & \longleftarrow &  & A
\end{matrix}
.\end{align*}

Barycentric Subdivision

Tables

Higher homotopy groups of {\mathbf{RP}}^n

Higher homotopy groups of {\mathbf{CP}}^n

Homotopy groups of spheres.

Homotopy groups of exceptional groups

Homotopy Groups of Lie Groups

  • \(O(n)\): \(\pi_k O_n = ?\)
  • \(U(n): \pi_k U_n\) is \({\mathbf{Z}}\) in odd degrees and \(\pi_1 U_n = 1\) 
    \todo[inline]{Check}
  • \(SU(n): \pi_k U_n\) is \({\mathbf{Z}}\) in odd degrees and \(\pi_1 U_n = 0\).
  • \(U_n: \pi_k(U_n)\) is \({\mathbf{Z}}/2{\mathbf{Z}}\) in degrees?

Higher Homotopy

  • \(n \geq 2 \implies \pi_n(X) \in \mathbf{Ab}\)

  • \(\Sigma S^n = S^{n+1}\)

  • \([\Sigma^n X, Y] \cong [X, \Omega^n Y]\)

  • \(\pi*n(\Omega X) = \pi*{n+1}(X)\)

    • \(\pi_n(X) \cong \pi_0(\Omega^n X)\)

  • \(n\geq 2 \implies \pi_n(S^1) = 0\)

  • \(k < n \implies \pi_k(S^n) = 0\)

  • \(\pi_n(X)\) is the obstruction to \(f: S^n \to X\) being lifted to \(\widehat{f}: D^{n+1} \to X\)

  • \(\pi_n(X) \cong H_n(X)\) for the first \(n\) such that \(\pi_n(X) \neq 0\); \(\forall k<n, ~H_k(X) = 0\).

  • \(k+2 \leq 2n \implies \pi_k(S^n) \cong \pi_{k+1}(S^{n+1})\)

  • \(\pi_k(S^n) = \pi_{k+1}S^{n+1} = \cdots =\pi_{k+i}S^{n+i}\)

  • \(F\to E \to B\) a fibration yields \(\cdots\pi_n(F) \to\pi_n(E) \to\pi_n(B) \to\pi*{n-1}(F) \cdots\)

  • Freundenthal suspension, stable homotopy groups

Higher Homotopy Groups of the Sphere

  • \(\pi_n(S^n) = {\mathbf{Z}}\)
  • \(\pi_{n+1}S^n = {\mathbf{Z}}_2\) for \(n \geq 4\)
  • \(\pi_{n+2}(S^n) \cong {\mathbf{Z}}_2\)
  • \(\pi_{n+3}S^n = {\mathbf{Z}}_8\) for \(n\geq 5\)
  • \(\pi_5 S^2 = {\mathbf{Z}}_2\)
  • \(\pi_6 S^3 = {\mathbf{Z}}_4\)
  • \(\pi_7 S^4 = {\mathbf{Z}}\oplus {\mathbf{Z}}_4\)
  • \(\pi_k S^2 \cong \pi_k S^3\)
  • \(\pi_3 S^2 \cong {\mathbf{Z}}\)
  • \(\pi_4 S^2 \cong {\mathbf{Z}}_2\)

Misc

  • \(\Omega({-})\) is an exact functor.

Building a Moore Space

  • To build a Moore space \(M(n, {\mathbf{Z}}_p)\), take \(X = S^n\) and attach \(e^{n+1}\) via a map \(\Phi: S^n = {\partial}B^{n+1}\to X^{(n)} = S^n\) of degree \(p\).
    • To obtain \(M(n, \prod G_i)\) take the corresponding \(\bigvee X_i\)
    • Can also use Mayer Vietoris to conclude \(H_{n+1}(\Sigma X) = H_n(X)\), and just suspend spaces with known homology.