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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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Boundary of a simplex

I've been wondering why the boundary of a simplex $\sigma : C_q (X) \rightarrow C_{q-1}(X)$ is defined to be $\partial \sigma = \sum (-1)^i \sigma \circ f_{i,q}$ with alternating sign. Why can it not be the sum over all faces without alternating…
Rudy the Reindeer
  • 46,359
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Compute homotopy classes of maps $[T^{2},T^{2}]$

How to compute $[T^{2},T^{2}]$ the set of homotopy class of continuous maps $f:T^{2}\longrightarrow T^{2}$? Thanks.
R.Bradley
  • 23
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Question about Lifting of Maps in the Circle

Let $S^1 = \{z\in\mathbb{C}:|z|=1\}$. For all $n\in\mathbb{N}$, define $f_n: S^1\to S^1$ by $f_n (z) = z^n$. Given $n\in\mathbb{N}$, for what values of $m\in \mathbb{N}$ there exists a lifting of $f_m$ to the recovering $f_n$?
user34870
  • 1,173
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union of two contractible spaces, having nonempty path-connected intersection, need not be contractible

show that union of two contractible spaces, having nonempty path-connected intersection, need not be contractible. can someone give me a proper example please.I could not remind anything.
daichi
  • 21
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to understand a theorem for fundamental group

I faced a problem to understand the proof of the following theorem from the book "algebraic topology by satya deo". If $F\colon X\to Y$ be a homotopy between two maps $ f,g\colon X\to Y $. Let $x_0\in X$ and $\sigma\colon I\to Y$ be the path…
hutom
  • 21
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Why Is the Induced Map Not Zero?

I am reading "Modern Classical Homotopy Theory" by Strom and have come across the following. We are given a fibration $F\rightarrow E\rightarrow B$. One then has two pushout squares: $$\require{AMScd} \begin{CD} F @>>> CF\\ @VVV @VVV \\ E @>>>…
J126
  • 17,451
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Homology groups (?) of some quotient of $\Bbb{R}P^n$

Here is a question from Hatcher (2.2.19): I assume that those $H_i$'s are homology groups, Hatcher denotes both the chain complexes and the homology groups by $H$ (I will denote the chain complexes by $C_i$). $\dots \xrightarrow{d_{n+1}}…
Xena
  • 3,853
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Homology of nonorientable surfaces

Let $N_g$ be a closed nonorientable surface of genus $g$. I will try to compute the homology groups and I want you to help me with certain steps and correct my mistakes - I will use this as an outline. (i) Cell decomposition: one 2-cell ($e^2$) g…
Xena
  • 3,853
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Rational Elienberg-Maclane Spaces

Is it true that $$ H^k(K(\mathbb{Z},n);\mathbb{Q})\cong H^k(K(\mathbb{Q},n);\mathbb{Q}) $$ for all $k$?
J126
  • 17,451
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The degree of antipodal map, composition of reflections?

Here is a bit from Hatcher's book: I don't understand part (f); why is the antipodal map the composition of $n+1$ reflections? Even if I accept that, I still don't know why does it have degree $(-1)^{n+1}$. What would that mean for $S^2$? We fix…
Xena
  • 3,853
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$S^n \rightarrow P^n$ the induced map on top cohomology class is zero.

How to show $f^*: H^n(RP^n;Z/2Z)\rightarrow H^n(S^n;Z/2Z) $ is zero map by definition? I know a proof by using the structure of cohomology ring of $P^n$, but the structure of the cohomology of $P^n$ seems to be hard to prove.
user93417
2
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1 answer

Two spaces with 'same' universal cover, must one cover the other?

The Klein bottle and the torus both have $\mathbf{R}^2$ as universal cover, and the torus can cover the Klein bottle. Does this always happen? If $A$ and $B$ have the same universal cover, must $A$ cover $B$ or vice versa? EDIT: as pointed out…
john w.
  • 653
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induced homomorphism on fundamental group

Let $S^1$ be the circle. Let $X= (S^1\times S^1)/(x,y)\sim (y,x)$. An element in $X$ is denoted $[x,y]$. Why the diagonal map $S^1\rightarrow X; x\mapsto [x,x]$ can not be surjective on $\pi_1$?
palio
  • 11,064
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1 answer

Hachter 1.1.14, prove on isomorphism by projection.

I am a bit confused with what I am suppose to prove here. I plan to go with prove isomorphism = homomorphism + bijection, but which function should I construct for the homomorphism? Show that the isomorphism $\pi_1( X \times Y) \approx \pi_1(X)…
1LiterTears
  • 4,572
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What is the meaning of concircular..

What is the meaning of the concircular? It says that the any five points are pairwise nonparallel and no four of them are concircular.
Ryan
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