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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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question about disconnected normal covering map

I have stuck on the problem in Hatcher, Algebraic topology, which claim that if the covering map $q\circ p:X\rightarrow Y \rightarrow Z$ is normal, then the covering $p:X\rightarrow Y$ also is normal. (Problem 16 in Section 1.3) The messy part of…
cjackal
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Finding when $\mathbb{C}P^n / \mathbb{C}P^{n-2}$ is homotopy equivalent to $S^{2n} \vee S^{2n-2}$ using Steenrod squares

Let $\mathbb{C}P^n$ denotes the complex projective space with real manifold dimension $2n.$ My question is for which values of $n$, the spaces $\mathbb{C}P^n / \mathbb{C}P^{n-2}$ is homotopy equivalent to $S^{2n} \vee S^{2n-2}?$ Attempt: The…
Math
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Degree of a $k$-fold covering projection is $k$

In Bredon's Topology and Geometry it‘s postulated that for a $k$-fold covering projection $p\colon X \to Y$, $\deg(p)=k$, where $\deg(p)=k$ The example given is of $X$ the $3$-dimensional sphere $S^3$ and $Y$ the Lens space $L(k,m)$, both of which…
x x
  • 143
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Why the attachment to simplices in (co)homology?

I've been thinking a bit about why we define the singular homology and cohomology groups with simplices rather than, say, cubes, and it seems to me that the elementary aspects of the theory would all become more elegant if we used cubes: Say we…
Noah Olander
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Lift of homeomorphism is homeomorphism

Suppose $p:Y \rightarrow X$ is the universal covering map of $X$. Given a contiunuous $f: X \rightarrow X$ then a well known theorem for existence of lifts states that there exist a continuous lift $$\tilde f : Y \rightarrow Y \text{ with } p \circ…
drunken_monkey
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First homology of a compact connected surface with boundary

I am looking for a practical description of the first homology group of $S_{g,b}$, the connected compact surface of genus $g$ with $b\geq 1$ boundary components. I think of $S_{g,b}$ as the $g$-holed torus with $b$ discs removed. I know that…
Bebop
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$[X,Y]$ is finite where $X$ is finite connected CW-complex, and $Y$ has finite homotopy groups

I have read this question in Allen Hatcher's book Algebraic Topology, (exercise 20, page 359): Show that $[X,Y]$ is finite if $X$ is a finite connected CW complex and $ \pi_i(Y) $ is finite for $ i \leq d:=\dim(X) $. I tried to use induction,…
HUO
  • 194
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Uniqueness of classifying space

Classifying spaces are obviously unique up to homotopy type. I am wondering, whether under stronger conditions, one can also say that they are unique up to homeomorphism. In particular, suppose $\Gamma$ is a group and there exist a model $X$ for…
Earthliŋ
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Does homotopy depend on function or the image?

I am just starting to read about algebraic topology, and I wonder whether homotopy depends on function or the image. According to Munkres' definition, two continuous function $f,g:[0,1]\to Y$ are said to be homotopic if there exists a continuous map…
Y.H. Chan
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Problem understanding how to compute fundamental group of connected sum of torus

I have attempted trying to compute the fundamental group of a 2 torus, however I don't know how to proceed to "simplify" the result after applying van Kampen's Theorem. I calculated the fundamental group of the torus $T$ to be $\pi_1 (S^1\times…
yoyostein
  • 19,608
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Spectral sequence for computing the homotopy fixed points in unstable equivariant homotopy theory

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question. Let $G$ be a topological group and $X$ be a $G$-space (for a nice notion of "space"). He defines $$X_{hG}=EG\times_G X$$ the homotopy orbit space,…
Bruno Stonek
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Why are higher homology groups not the abelianizations of higher homotopy groups?

Really the question is exactly the title: Why (conceptually and geometrically if possible) are higher homology groups not the abelianizations of higher homotopy groups?
user153312
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Map from $n$-sphere to $n$ dimensional torus

Let $n\ge 2$. How can you prove that for every continuous $f:S^n\to T^n$, the induced map on singular homology $f_\star:H_n(S^n)\to H_n(T^n)$ is the zero map? Here, $S^n$ is the $n$ dimensional sphere, and $T^n=(S^1)^n$ is the $n$ dimensional…
Mike Earnest
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Prove that $S^{n-1} \simeq \Bbb R^n - \{ 0 \}$.

That is prove that the $n-1$ sphere is homotopic to the Euclidean space without the origin. Two topological spaces $X, Y$ are said to be homotopic if there are maps $f: X \to Y$ and $g: Y \to X $ such that $f \circ g \simeq Id_Y$ and $g \circ f…
Lemon
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Is there an isomorphism $\mathrm{Hom}(H_1(X),G)\simeq\mathrm{Hom}(\pi_1(X),G)$ when $X$ is path connected?

In Hatcher 3.1.5 on pg. 205, one proves that if $\varphi\in C^1(X;G)$ is a cocycle, where $X$ a space and $G$ an abelian group, then for paths $f$ and $g$ one has various properties $\varphi(f\cdot g)=\varphi(f)+\varphi(g)$, $\varphi$ sends constant…
Ghost Beach
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