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In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48): "[Bott] taught many of us to think functorially, like thin …

An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, …

I was inspired by the following algebraic topology orals question: "Is $S^1$ the loop space of another space?" This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop sp …

I just revamped what was written. Perhaps now it is more understandable:New Wiki Entry on Verdier duality.

I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold. I believe that some sort of surgery will do the trick. Roughly speaking, I w …

This question loosely elaborates on an earlier question. It is pretty silly, but I'd like to hear some authoritative answers. Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of s …

Cosheaves are indeed mysterious gadgets. On the one hand, cosheaves are everywhere, but on the other hand, someone used to thinking sheaf-theoretically may have some problems. I am very close to finis …

Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$. There are many different categories that one can asso …

Also I ask, are there any additional ways to define orientability/orientation for a differentiable manifold(not just for a vector space)? Another notion of orientability is the existence of an at …

There are lots of possible answers to your question, but maybe here are some ideas. They aren't papers, but good projects. Method of Characteristics in First Order Nonlinear PDE can be interpreted v …

This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 - Non-abelian moduli spaces). Here he says that for a Riemann surface $\Sigma$ the first …

Here are two comments: 1) I suspect the answer is "yes," so long as your homotopy has the property that your pullback is locally constant (and hence, by triviality of the interval, constant) along th …