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 that the based loop space $\Omega X$ of some topological space $X$ is an $A_\infty$ space. Moreover, Stasheff proved that any $A_\infty$ space is homotopic to $\Omega X$. Stasheff also defined maps between $A_n$ spaces as $A_n$ maps, which are the right notion of morphism in the category of $A_n$ spaces.
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 $A_n$ spaces. That is, whether there exists a homotopy equivalence $f:\Omega B\times \Omega F\to \Omega E$, which can be extended to an $A_n$ map. So far, we are only able to find the condition in terms of brace product (introduced by I. M. James) in a fibration with section in the case of $n=2$.
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.
Abstract: Given a fibration $F \hookrightarrow E \rightarrow B$ with a homotopy section $s : B \rightarrow E$, James introduced a binary product $\left\{ , \right\}_s : \pi_i B \times \pi_j F \rightarrow \pi_{i+j-1} F$, called the brace product. In this article, we generalize this to general homotopy groups. We show that the vanishing of this generalized brace product is the precise obstruction to the $H$-splitting of the loop space fibration, i.e., $\Omega E \simeq \Omega B \times \Omega F$ as $H$-spaces. Using rational homotopy theory, we show that for rational spaces, the vanishing of the generalized brace product coincides with the vanishing of the classical James brace product, enabling us to perform relevant computations. In addition, the notion of $J$-homomorphism is generalized and connected to the generalized brace product. Among applications, we characterize the homotopy types of certain fibrations including sphere bundles over spheres.