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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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21356 questions
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Empty faces in $\Delta$-complexes

Although Hatcher calls the empty set a simplex in his book Algebraic topology on page 110 he seems to assume that $0$-simplices $[v_0]$ have no faces at all in the definition of $\Delta$-complexes on page 102. Therefore, the condition that the…
Jochen
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Relative homology and homology of the quotient

Let $X$ be a topological space. Then, the cone $CX$ over $X$ is always contractible. Hence, $CX \simeq \{p\}$, and $$ H_n(CX) = \begin{cases} \mathbb{Z}, &\quad n = 0,\\ 0, &\quad n \neq 0. \end{cases} $$ On the other hand, $X \cong X \times \{0\}…
warzasch
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Applications of Excision and Computations of Homology Groups

I was wondering where exactly we are using Excision. I read the relation between relative homology and quotient homology recently. It says something like it can be applied on good pairs. I am looking for examples of good pairs upon which I can…
permutation_matrix
  • 1,356
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absence of global orientation for a non compact manifold

Massey Singular Homology theory states at IX.2 Let $M$ be an n-dimensional manifold with orientation $\mu$; it would be advantageous if there were a global homology class $\mu_M$ \in $H_n(M,\mathbb{Z})$ such that for any $x \in M$ $\mu_x =…
namenick
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Fundamental Group

Consider some space $X$. Does the fundamental group tell us information about the equivalence between two paths $f,g: I \to X$? So there exists a homotopy $h: I \times I \to X$ such that $h(s,0) = f(s)$, $h(s,1) = g(s)$, $h(0,t) = x$ and $h(1,t) =…
Damien
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The induced map of a continuous map $F\colon X\to X$ homotopic to the identity is an isomorphism

In some lecture notes I am reading through there is this result and a proof of it that I don't seem to understand. Lemma: If $F:X→X$ is a continuous map which is homotopic to the identity map $id_X:X→X$, then the induced map…
Mafematician
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A simpler way to understand the isomorphism: $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$?

I'm trying to figure out the below sentence (in bold, starting with "since.." from Hatcher formally: ...This is done by defining the reduced homology groups $\tilde{H}_n(X)$ to be the homology groups of the augmented chain complex $$ \cdots \to…
Anon
  • 1,757
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Mayer-Vietoris for contractible intersections

Given a countable union of spaces $$X=\bigcup_n Y_n$$ such that all intersections $$\bigcap_{i\in I}Y_i, \vert I\vert\ge 2$$ are contractible (or weaker just have trivial homology). Is it true that in homology (in degrees $*\ge…
user39082
  • 2,124
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Why do we need to quotient out by the normal subgroup in Van-Kampen's theorem?

I've found few different formulations for Van-Kampen's theorem the first of which states that if $X = A \cup B$ where $A$ and $B$ are path-connected, contain the basepoint $x_0$ and $A \cap B$ is path-connected then $$\pi_1(X) \cong \pi_1(A) \ast…
Emil Grönberg
  • 277
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Detection of the composition $\eta^2$ of the Hopf map $\eta$

Let $\eta=\eta_n\colon S^{n+1}\to S^n$ be the iterated suspension of the Hopf map $\eta\colon S^3\to S^2$. It is known that $\eta$ is detected by the Steenrod square for $n\geq 3$. Is there any (cohomology) operations capable of detecting the…
LipCaty
  • 379
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Proving the Ham-Sandwich theorem for $n=3$

Let $A_1, A_2, A_3$ be compact sets in $\Bbb R^3$. Use the Borsuk–Ulam theorem to show that there is one plane $P \subset \Bbb R^3$ that simultaneously divides each $A_i$ into two pieces of equal measure. Every point $s \in \Bbb S^2$ defines a…
Emil Grönberg
  • 277
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Fundamental group of the quotient of $\partial I$ with a simply connected space

I was wondering under what conditions will the quotient of I at its endpoints $\partial I$ with a simply connected space will give $\pi_1 = \mathbb{Z}$. To be clear, the end points do not need to quotient to the same point. I think I have a proof…
adumbdumb
  • 21
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Definition on homotopy of maps of pairs and deformation retract of pairs

Reading hatcher's algebraic topology, there is a statement(page127): Proposition 2.19. If two maps $f, g: (X, A) \rightarrow (Y, B)$ are homotopic through maps of pairs $(X, A) \rightarrow (Y, B)$, then $f_* = g_* : H_n(X, A) \rightarrow H_n(Y,…
onRiv
  • 1,294
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Why using closure of U to define the topology on the orientation bundle of a topological manifold?

Sorry I have to quote a major part of content from Bredon’s /Topology and Geometry/ as a screenshot. In the following definition of the basis $U_\alpha$, why using the closure of $U$ but not $U$ directly? It seems the arguments for $U_\alpha$ being…
onRiv
  • 1,294
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Universal coefficient theorem

Let $M$ be an $R$-module where $R$ is a P.I.D. we have the exact sequence $$0\rightarrow \operatorname{Ext}_R(H_{q−1}(X;R),M) \rightarrow H^q(X;M) \rightarrow \operatorname{Hom}(H_q(X;R),M)\rightarrow 0$$ Is $H_q(X;R)$ an $R$-module? and then…
palio
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