My broad area of research is algebraic topology. More specifically, I am working in homological algebra and homotopy theory. I am also interested in differential topology & geometry, algebraic geometry.
Suppose $X$ be a topological space with a binary operation i.e., there is a map from $X\times X\to X$ taking $(a,b)\mapsto ab$. Now if the
binary operation is not associative, we have two non-identical cannonical maps from $X\times X\times X\to X$ namely $f_{1,2}:(x_1,x_2,x_3)\mapsto x_1(x_2x_3)$
and $f_{2,1}:(x_1,x_2,x_3)\mapsto (x_1x_2)x_3$.
$f_{1,2}$ and $f_{2,1}$ are not equal but we ask whether they are homotopic. If homotopy associative (called weakly associative), then we
have a map $m_3:K_3\times X^3\to X$ defined through the homotopy between $m\circ (m\times 1)$ and $m\circ (1\times m)$, where $K_3$ is
an interval. Similarly, we can define five different maps from $X^4\to X$ using $m$, and between any two such maps, there are two
different homotopies (using the chosen homotopy associativity). If those two homotopies are homotopic themselves, then this defines a
map $m_4:K_4\times X^4\to X$, where $K_4$ is a filled pentagon. If we continue this process, we obtain a map $m_n:K_n\times X^n\to X$
for $n\geq 2$. These complexes $K_n$, called associahedra, are used as a parameter space to define $A_n$ spaces (see Homotopy Associativity of H Spaces by J. D. Stasheff for more details). Any associative $H$-space is an $A_n$ space for any $n (\geq 2)\in \mathbb{N}$. If a space is $A_n$ space for any $n(\geq 2)\in \mathbb{N}$, it is called $A_\infty$ space. It is known (due to Stasheff) that any $A_\infty$ space is homotopic to $\Omega X$, based loop space of some topological space $X$. Stasheff also defined maps between $A_n$ spaces as $A_n$ maps.
In this regard, we are interested in the loop space fibration $\Omega F\hookrightarrow \Omega E\rightarrow \Omega B$ of any given fibration $F\hookrightarrow E\rightarrow B$. We are investigating the possible algebraic/topological conditions which ensure $\Omega B\times \Omega F$ and $\Omega E$ are isomorphic as $A_\infty$ spaces.
Abstract: We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the study of $A_\infty$ maps. We also provide new combinatorial isomorphisms between Stasheff associahedra, collapsed multiplihedra, and graph cubeahedra for path graphs.