If you really must use a list environment in your abstract, I suggest using the enumitem package to manage the horizontal spacing or margins. Note that the output from your code snippet produces results as expected, since the "first paragraph" (starting with "We discuss...") is actually indented by a length \parindent. Either way, here is a take on a possibility that you might be after:

\documentclass{article}
\usepackage{showframe}% http://ctan.org/pkg/showframe
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\usepackage{enumitem}% http://ctan.org/pkg/enumitem
\begin{document}
\begin{abstract}
\lipsum[1]
We discuss several exciting topics:
\begin{enumerate}[label={Topic~\arabic*:},leftmargin=*]
\item \lipsum[2]
\item \lipsum[3]
\end{enumerate}
And here we have the end of the abstract.
\end{abstract}
\end{document}
Some more things to note:
- Since
\lipsum typesets paragraphs, it issues \par at the end of every paragraph. This makes the following paragraph indent by \parindent, as you can see from the MWE.
- Since there is no gap between the end of the
enumerate environment and the start of the last sentence/paragraph, no \parindent is visible. If you want a paragraph indent, issue \par at the end of enumerate, or use \indent.
- I've made the list enumerate itself as
Topic~\arabic*: by using the functionality of enumitem's label setting.
- Also, making the left margin flush with the text block (in
abstract), I've used leftmargin=*. Other alignment options are also available.
showframe was used to highlight the text block margins, while lipsum provided some dummy text.
In an attempt to replicate your abstract without any package dependencies, consider the following:

\documentclass{article}
\begin{document}
\begin{abstract}
This paper studies the one-way communication complexity of the \textit{subgroup
membership problem\/}, a classical problem closely related to basic questions in quantum
computing. Here Alice receives, as input, a subgroup~$H$ of a finite group~$G$; Bob receives
an element $y\in G$. Alice is permitted to send a single message to Bob, after which he must
decide if his input~$y$ is an element of~$H$. We establish the following bounds on the classical
communication complexity of this problem in the bounded-error setting: \par \medskip
\addtolength{\leftskip}{1.5em} \setlength{\parindent}{-1.5em}
\makebox[1.5em][l]{1.}The problem can be solved with $\mathcal{O}(\log|G|)$-bit communication with the promise that~$H$ is normal. \par \medskip
\makebox[1.5em][l]{2.}The problem can be solved with $\mathcal{O}(d_{\max}-\log|G|)$-bit communication, where~$d_{\max}$ is the maximum
degree of an irreducible complex representation of~$G$. \par \medskip
\makebox[1.5em][l]{3.}For any prime~$p$ not dividing~$|G|$, the problem can be solved with $\mathcal{O}(\log|G|+d_{\max}\cdot\log p)$-bit
communication, where~$d_{\max}$ is the maximum degree of an irreducible $\mathbf{F}_p$-representation of~$G$.
\end{abstract}
\end{document}
This way you typeset an enumeration by physically adjusting the left margin (\leftskip) and accompanying paragraph indent (\parindent) to simulate enumerate. I've hardcoded the numbering, but that can be made into a counter as well. However, this would only be necessary if they may be re-ordered often, or if you want to reference them by means of \label and \ref commands. Also, I've placed a \medskip between the items to make it look like an enumeration. However, a single \par would also suffice.
enumerateis present.amsartis one such. perhaps you could crib some code from that. (i haven't explored to see how difficult it might be.) i think the problem was addressed there because sometimes numbered displays are embedded in the abstract, and the margins have to be honored for those. – barbara beeton Dec 22 '11 at 18:05abstractenvironment (or code), but rather with yourenumeratecode. Without the actual class code, it would be hard to assess how to fix your current situation. – Werner Dec 24 '11 at 4:41