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    "# Steenrod squares\n",
    "\n",
    "\n",
    "In 1947, Steenrod introduced by means of formulae the cup-$i$ coproducts on the cochains of spaces. These bilinear maps give rise to the natural cohomology operations \n",
    "\\begin{equation*}\n",
    "Sq^k \\colon H^\\bullet(X; \\mathbb F_2) \\to H^\\bullet(X; \\mathbb F_2)\n",
    "\\end{equation*} \n",
    "\n",
    "going beyond betti numbers and laying at the heart of homotopy theory.\n",
    "\n",
    "As motivation for the practical use of Steenrod squares $Sq^k$ here are a few situations where they are a convenient distinguishing tool.\n",
    "\n",
    "1. The real projective plane (sphere with antipodal points identified) and the wedge of a circle and a sphere (their union with a point from each identified together) have, with $\\mathbb F_2$-coefficients, the same betti numbers, yet the rank of $Sq^1$ tells them apart.\n",
    "\n",
    "\n",
    "2. Similarly, the complex projective plane and the wedge of a 2-sphere and a 4-sphere have the same betti numbers with any coefficients, yet the rank of $Sq^2$ tells them apart.\n",
    "\n",
    "\n",
    "3. The suspensions (join with two points) of the two spaces above have the same betti numbers and also isomorphic cohomology rings (cup product), yet $Sq^2$ tells them apart.\n",
    "\n",
    "The cup-$i$ coproducts $\\Delta_i \\colon C_\\bullet(X; \\mathbb F_2) \\to C_\\bullet(X; \\mathbb F_2) \\otimes C_\\bullet(X; \\mathbb F_2)$ extend the Alexander-Whitney diagonal $\\Delta_0$, and the higher cup-$i$ coproducts can be interpreted as coherent homotopies witnessing at the chain level the symmetry of $\\Delta_0$ at the homology level.\n",
    "The Steenrod squares are defined by\n",
    "\n",
    "\\begin{equation}\n",
    "Sq^k \\big( [\\alpha] \\big) = \\big[ (\\alpha \\otimes \\alpha) \\Delta_{|\\alpha|-k}\\big].\n",
    "\\end{equation}\n",
    "\n",
    "Let us review their description as presented in [MM22a] proven to be equivalent to Steenrod's original in [MM22b].\n",
    "\n",
    "Let $X$ be a simplicial complex and $x \\in X_n$.\n",
    "For a set\n",
    "\\begin{equation*}\n",
    "U = \\{u_1 < \\dots < u_r\\} \\subseteq \\{0, \\dots, n\\}\n",
    "\\end{equation*}\n",
    "\n",
    "we write $d_U(x) = d_{u_1}\\! \\dotsm \\, d_{u_r}(x)$, with $d_{\\emptyset}(x) = x$ and $d_u [v_0, \\dots, v_n] = [v_0, \\dots, \\widehat v_u, \\dots, v_n]$.\n",
    "\n",
    "For any simplicial complex $X$ and integer $i$\n",
    "\\begin{equation*}\n",
    "\\Delta_i \\colon C_\\bullet(X; \\mathbb F_2) \\to C_\\bullet(X; \\mathbb F_2) \\otimes C_\\bullet(X; \\mathbb F_2)\n",
    "\\end{equation*}\n",
    "\n",
    "is the linear map defined on a simplex $x \\in X_n$ to be $0$ if $i \\not\\in \\{0, \\dots, n\\}$ and is otherwise given by\n",
    "\n",
    "\\begin{equation}\n",
    "\\Delta_i(x) = \\sum d_{U^0}(x) \\otimes d_{U^1}(x)\n",
    "\\end{equation}\n",
    "\n",
    "where the sum is taken over all subsets $U = \\{u_1 < \\cdots < u_{n-i}\\} \\subseteq \\{0, \\dots, n\\}$ and\n",
    "\n",
    "\\begin{equation} \\label{e:partition subsets}\n",
    "U^0 = \\{u_j \\in U\\mid u_j \\equiv j \\text{ mod } 2\\}, \\qquad\n",
    "U^1 = \\{u_j \\in U\\mid u_j \\not\\equiv j \\text{ mod } 2\\}.\n",
    "\\end{equation}"
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   "source": [
    "from itertools import combinations\n",
    "\n",
    "def steenrod_diagonal(i, spx):\n",
    "    '''Returns the image of applying the ith Steenrod coproduct to spx.'''\n",
    "    answer = []\n",
    "    n = len(spx)-1\n",
    "    for U in combinations(range(n+1), n-i):\n",
    "        left, right = list(spx), list(spx)\n",
    "        for u in U:\n",
    "            if (U.index(u) + u) % 2 == 1:\n",
    "                left.remove(spx[u])\n",
    "            else:\n",
    "                right.remove(spx[u])\n",
    "        answer.append((tuple(left), tuple(right)))   \n",
    "    return answer"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Let us consider a simplicial complex modeling the real projective plane $\\mathbb RP^2$ whose cohomology is (additively) given by\n",
    "\n",
    "\\begin{equation}\n",
    "H^n(\\mathbb RP^2; \\mathbb F_2) \\cong\n",
    "\\begin{cases}\n",
    "\\mathbb F_2, & n=0,1,2 \\\\\n",
    "0, & \\text{otherwise}.\n",
    "\\end{cases}\n",
    "\\end{equation}"
   ]
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   "source": [
    "rp2 = (\n",
    "    (0,), \n",
    "    (1,), (0,1), \n",
    "    (2,), (0,2), (1,2), (0,1,2),\n",
    "    (3,), (0,3), (1,3), (0,1,3), (2,3),\n",
    "    (4,), (0,4), (1,4), (2,4), (1,2,4), (3,4), (0,3,4), (2,3,4),\n",
    "    (5,), (0,5), (1,5), (2,5), (0,2,5), (3,5), (1,3,5), (2,3,5), (4,5), (0,4,5), (1,4,5)\n",
    "    )"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As computed in the notebook *barcode*, a cocycle representative $\\alpha$ for the generator of $H^n(\\mathbb RP^2; \\mathbb F_2)$ is the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
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   "source": [
    "coho_rep = [(2, 4), (1, 5), (1, 4), (2, 3), (3, 5)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now use the linearity of the tensor product to compute by brute force a representative of $Sq^1\\big( [\\alpha] \\big)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(2, 3, 5)]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from itertools import permutations\n",
    "\n",
    "sq_rep = []\n",
    "for pair in permutations(coho_rep, 2):\n",
    "    for spx in (s for s in rp2 if len(s) == 3):\n",
    "        if pair in steenrod_diagonal(0, spx):\n",
    "            sq_rep.append(spx)\n",
    "\n",
    "sq_rep"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since this cocycle represents the generator of $H^2(\\mathbb RP^2; \\mathbb F_2)$, we have that $\\mathrm{rank}(Sq^1) = 1$."
   ]
  }
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