Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …

In 1968, Arnol'd proved that the integral cohomology of the pure braid group $P_n$ is isomorphic to the exterior algebra generated by the collection of degree-one classes $\omega_{ …

I'm wondering if a compact set $A\subset\mathbb{C}$ satisfying the properties that • $A$ and its complement have finitely many connected components • every connected component of …

I'm reading a paper On the Torsion Jacquet-Langlands correspondence by Akshay Venkatesh and Frank Calegari. Truthfully speaking I have no idea what Jacquet-Landlands is. I'm ju …

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category …

What is a motivation to study topological conjugacy of a real-valued functions on a manifold? (The importance of notion of a topologically conjugate homeomorphisms is clear for me) …

I would like to know the ring structure of $K(Q_n)$ explicitly where $Q_n \subset \mathbb{P}^{n+1}$ is the non-singular $n$-dimensional complex quadric and $K(Q_n) = K^0(Q_n)$ is …

As pointed out by David White in http://mathoverflow.net/questions/73687/when-mapping-cone-is-contractible there exists an acyclic CW-complex $X$ which is not contractible but who …

In his book Quantum invariants of knots and 3-manifolds page 124, Turaev defined a TQFT $\tau$ axiomatically. For a cobordism $(M, \partial_{-}M, \partial_{+}M)$, a TQFT assignes …

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able …

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies. DEFINITION   A continuous map &n …

Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions. Thanks to Milnor, we know that the $\pi_1$ any compact …

Does every connected metrizable locally path connected topological space X admit a compatible metric d so that (X,d) is a length space? (Recall the metric space (X,d) is a length …

Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with orient …

Suppose I have $X=X_1\cup X_2\cup…\cup X_n$ and $f:X \to Y$ where $Y$ has a similar decomposition. Suppose I know that $f | X_{i_1}\cap…\cap X_{i_r} \to Y_{i_1}\cap…\cap Y_{i_r}$ …