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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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fundamental group and admitting a z_2-map to circle

There is a lemma (Lemma 6.1) in this paper https://arxiv.org/pdf/1710.05290.pdf without proof. I would be so grateful if someone could help me regrading the proof. For completeness, I rewrite the lemma here: Notation: In following, the orbit space…
123...
  • 959
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Prove that every continuous map $f: S^1 \to S^1$ is homotopic to a continuous map $g: S^1 \to S^1$ with $g(1) = 1$

I am asked to prove that every continuous map $f: S^1 \to S^1$ is homotopic to a continuous map $g: S^1 \to S^1$ with $g(1) = 1$. Now I thought intuitively I could somehow take each $f(x,y)$ at time $t = 0$ and just go around the circle to get to…
HAT
  • 285
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Homology class of sphere in projective space

Let $\mathbb{F}_2$ be the field with two elements. Then consider the homology of the real projective space with $\mathbb{F}_2$ coefficients $H_k(\mathbb{RP}^n; \mathbb{F}_2)\cong\mathbb{F}_2$ for $0< k< n$. I wonder which homology classes can be…
Hans
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How to prove that this space cannot be embedded in $\mathbb{R}^3$?

Take two Moebius bands glued together as in the picture here. Now attach a $2$-cell along one of the boundary components of this space (indicated in green in the picture). Call the resulting space $X$. My gut feeling is that $X$ cannot be embedded…
a student
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3 answers

rational homology

what's the isomorphism between $H_*(X;\mathbb Q)$ and $ H_*(X;\mathbb Z)\otimes \mathbb Q$
studento
  • 155
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A basic reduced homology isomorphism

How do you prove the isomorphism $\tilde{H}_{n}(X/A,A/A)\cong\tilde{H}_{n}(X/A)$ for reduced homology groups? Is it just a matter of seeing that chains into the point $A/A$ end up being trivial in the latter group?
SihOASHoihd
  • 1,886
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Operation on spaces and impact on algebraic invariants

I'm trying to get a feel for why different operations on spaces are useful. I realize this question is very long if someone wants to give a response to all the cases. With ''operations on spaces'' I mean: product, wedge product, cone, suspension,…
dstt
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Does strong deformation retraction move points along fixed x curve along that curve?

Let $X$ be a topological space, and $f : X \times [0, 1] \to X$ be a strong deformation retraction from $X$ to $X' \subset X$. Does it hold that for each $x \in X$ and $t \in [0, 1]$, $$f(f(x, t), 1) = f(x, 1)$$ ? This property does not hold for…
kaba
  • 2,035
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Covering map Problem and induced isomorphism

I don't know how to start the following question. Any help will be appreciated! Thank you! Suppose that $S^1 \times P^2$ covers some space, and let $h$ be a covering translation. Show that the induced isomorphism $h$ of $H_1(S^1 \times P^2)$ must…
Susan
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Do Homotopy Equivalent, Orientable n-Manifolds Have The Same Cohomology With Compact Support?

Using Poincare duality, I believe that if $X$ and $Y$ are two orientable $n$-manifolds with $X\simeq Y$, then we should have $H^i_c(X)\cong H_{n-i}(X)\cong H_{n-i}(Y)\cong H^i_c(Y)$ for all $i$. However, in a comment to the question linked at the…
Alex Menzia
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1 answer

Is connectivity of a spectrum $E$ representing a cohomology theory equivalent to $E^{>1}(*)=0$?

Let $E$ be a spectrum, $X$ a CW-complex and set $$ E^k(X)=[X_+,S^k\wedge E] $$ for an integer $k$ as the $k$-th cohomology group of $X$ associated to the spectrum where the brackets denote stable homotopy classes and where I suppress the symbol…
Ronald Bernard
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On a Remark on the van Kampen theorem in Rolfsen's Knots and Links

In Appendix A, p. 372 of Rolfen's 'Knots and Links', Rolfsen states the van Kampen theorem, and follows with: "Somewhat more generally, suppose the inclusion homomorphisms $i_{1^\ast}$ and $i_{2^\ast}$ are injective, then one may deduce that…
Dario
  • 73
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Are the homology groups for simplicial homology and singular homology with coeeficients the same?

Let $X$ be a topological space with a $\Delta$-complex structure. We know that $H_n^\Delta(X)\cong H_n(x)$ (Theorem 2.27 Algebraic Topology Hatcher). My question is the following: Does this also hold for other coefficients than $\mathbb{Z}$? To be…
Frederik
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Proof of the formula $\chi=2-2g$ for closed orientable surfaces

Is there any simple proof for the formula $\chi=2-2g$, where $\chi$ is the Euler characteristic and $g$ the genus? This is likely to be answered before, but I cannot find it. I use the definitions $\chi = \sum (-1)^k \text{rk} (H_k(X;\mathbb Z))$…
Ma Joad
  • 7,420
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1 answer

Correlation between winding number and degree.

Let $\mathbb{C}^*=\mathbb{C} \setminus\{0\}$. Suppose $f:S^1\rightarrow\mathbb{C}^*$ is a continuous map such that the composite function $\alpha=f\circ\pi:[0,1]\rightarrow\mathbb{C}^*$ is piecewise continuously differentiable. The winding number of…
Walt
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