italic after math equation: how to solve (not solved with $)

Question

I am not an expert with equations on latex. I have wrote these ones within a chapter of a book, but I missed how to write the label on the left with the equation number. Moreover, the text after the equations is italic for a couples of lines and I did not find the way to fix it. What is wrong?

(Edit with suggestions proposed)

\documentclass[a4paper,11pt,  twoside, openright]{book}
\usepackage[T1]{fontenc}                
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[a4paper,top=3.5cm,bottom=3cm,left=3cm,right=3cm]{geometry}
\usepackage{fancyhdr}
\usepackage{setspace}
\usepackage{graphicx}
\usepackage{caption}
\captionsetup{font=small}
\usepackage{topfront}
\usepackage{emptypage}
\usepackage{natbib, authoryear}
\graphicspath{figure}
\usepackage{longtable}
\usepackage{chngcntr}
\usepackage{rotating}
\usepackage{mathtools}
\usepackage{textgreek}
\usepackage{hyperref}

\begin{document}

yi \in Y_{i} \sim mathrm{Multinom}(\pi^1_i , \pi^2_i , \pi^3_i ) 
\pi^k_i = \frac{exp(\eta^k_i)} {1+ \exp(\eta^2i) + exp(\eta^3i)}\hfill     k= 1,...,3

\eta^k_i = ln(\frac{\pi^k_i }{\pi^1_i}) = \beta^k_0 + x_i_1\beta^k_1 +    \sum^11_{j_=_3 } x_i_j\beta^k_j \hfill k= 2,3
\eta^1_i = 0 

some text which is italic
\end{document}

i have not solved the issue substituing with $\pi$

Answer 1

I had to take some guesses as to the intended meaning in some cases...

\documentclass[a4paper,11pt,  twoside, openright]{book}
\usepackage[T1]{fontenc}                
\usepackage[utf8]{inputenc}

\usepackage{mathtools}

\DeclareMathOperator\Multinom{Multinom}

\begin{document}

some text
\begin{equation}
yi \in Y_{i} \sim \Multinom(\pi^1_i , \pi^2_i , \pi^3_i ) 
\end{equation}

some more text ${x_i}_1$ ?? $\sum^1$ ???
\begin{align}
\pi^k_i &= \frac{\exp(\eta^k_i)} {1+ \exp(\eta^2i) + \exp(\eta^3i)}  &&  k= 1,\dots,3\\
\eta^k_i &= \ln(\frac{\pi^k_i }{\pi^1_i}) = \beta^k_0 + x_{i1}\beta^k_1 +    \sum^11_{j=3 } x_{ij}\beta^k_j &&k= 2,3\\
\eta^1_i &= 0 
\end{align}

some text which is italic
\end{document}