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Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Let us denote $G=\pi_1(X,x_0)$ and $H=p_*(\pi_1(\bar{X},\bar{x}_0)$. The first statement, about the cyclicity of the group $\text{deck}(X,p)$ of deck transformations, follows when you observe that $\ … answered Jan 13 '15 by Espen Nielsen 0answers I have to show that if$X$is a CW-complex,$A$is a subcomplex and$i:A\hookrightarrow X$is a cellular inclusion, then$i$is a cofibration. My attempt is as follows. I think my proof, which first p … asked Jan 24 '13 by Espen Nielsen We talk about lifting a map$f$through another map$\rho$. The map$\rho$is therefore a part of the data, together with the spaces$X,\tilde{X}$and the map$f$. answered May 5 '14 by Espen Nielsen Given a smooth curve$\gamma:I\rightarrow \mathbb{R}^2$, You can define the Gauss map$n:I\rightarrow S^1\subset \mathbb{R}^2$giving you a unit vector normal to your curve at every point in a smooth … answered Nov 2 '12 by Espen Nielsen I will show you that whevener you have chosen a homotopy$H$, you can construct a new homotopy which is constant on$S^{n-1}$. First, we label points on$D^n$as$(r,\theta)$, treating$r$as a cont … answered Jan 14 '15 by Espen Nielsen 2answers I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books. One part is standard: Suppose there is an$f:D^n \rightarrow D^n$with no fixed poin … asked Nov 30 '12 by Espen Nielsen 3answers It is easy to show that for any topological space$X$, the cone$CX$is contractible. I am interested in the converse. If$Y$is a contractible space, is$Y$homeomorphic to$CX$for some topological … asked Jan 16 '13 by Espen Nielsen If you have two continuous maps$f:Z\rightarrow X$and$g:Z\rightarrow Y$, the adjunction space (or pushout, as it is usually called), is the space$W=X\coprod_ZY=(X\coprod Y)/\sim$, where$\sim$is t … answered Nov 2 '14 by Espen Nielsen 1answer So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not … asked Jan 22 '13 by Espen Nielsen For part 1: The part where you (between lines) write$H_{n-1} \circ \partial '_n \circ i_n = -h_{n-2}\circ \partial^X_{n-1}$is the error. Remember that$\partial '(0,0,z)=(-z,z,-\partial^X z)\$. Then …
answered Feb 4 '15 by Espen Nielsen
For a quick introduction, you can read this AMS survey. What is … Persistent Homology? by Shmuel Weinberger A basic notion in persistent homology is a barcode. The following article gives an intro …
answered Oct 15 '15 by Espen Nielsen