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May
3
answered Pull-back of a fibration along a homotopy equivalence
May
2
revised Visualizing products of $CW$ complexes
typo
May
2
revised Visualizing products of $CW$ complexes
added an account of a pyjama trick
May
2
revised Visualizing products of $CW$ complexes
slight clarification and a correction to 1/2 at the end
May
2
answered Visualizing products of $CW$ complexes
May
1
comment How to compute a homotopy to show the operation on the fundamental group is assoicative?
This is an interesting exercise in linearity, but I have not heard it explained why so many (most?) topology texts insist that paths have to be "of length 1", i.e. maps $f:[0,1]\to X$, rather than of length $r \geqslant 0$, i,e, maps $[0,r] \to X$. With the more general definition, the paths in a space form a category under composition. You still have to do something on reparametrisation, and a notion of equivalence. Almost all texts also avoid the fundamental groupoid on a set of base points,.
Apr
30
revised An equivalence of categories
added the worh"uniquely", and a modification
Apr
30
answered An equivalence of categories
Apr
30
answered The Plate Trick and $SO(3)$
Apr
27
answered relative homotopy groups
Apr
26
revised Prove that exist bijection between inverse image of covering space
added another link
Apr
26
answered Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?
Apr
25
answered Prove that exist bijection between inverse image of covering space
Apr
23
answered simply connected covering of a path connected space (II)
Apr
23
answered $\pi_1$ and $H_1$ of Symmetric Product of surfaces
Apr
22
comment Finding the Fundamental Groups of Some Modular Spaces
An older version of this chapter on orbit grouopids is available at . arXiv:math/0212271.
Apr
22
comment Finding the Fundamental Groups of Some Modular Spaces
@QiaochuYuan: Chapter 11 of my book Topology and Groupoids is on orbit spaces and orbit groupoids. In particular, it proves that for a discontinuous action on a Hausdorff space which is locally nice, then the fundamental groupoid of the orbit space is isomorphic to the orbit groupoid of the fundamental groupoid. The one calculation given is that of the fundamental group of the symmetric square of a space. So I am hoping for more applications!
Apr
20
comment Simple homotopy construction
This matter is related to my answer to math.stackexchange.com/questions/1237690/…; why do most texts insist that paths and loops have to be maps from the unit interval $[0,1]$?
Apr
18
answered Homology and (co)Limits
Apr
17
revised On the associative property of a binary operation of the fundamental group.
slight correction to a quotation