Cohomology ring of RPn\mathbb RP^n with integral coefficient.

I know cup product structure on H∗(RPn;Z2)=Z2[α]/(αn+1). How to get H∗(RPn;Z) from this? I have two cochain complexes for two coefficient rings. Now my question is what will be the induced map between these two cochain complexes and what will be H∗(RPn;Z)? Answer I think, H∗(RP2n+1)=Z[α,β]/(2α,αn+1,αβ,β2), where α has degree 2 and β has degree … Read more

$M$ not orientable implying results about $H_{n-1}(M, \mathbb{Z})$, $H_n(M, \mathbb{Z}_q)$?

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. How do I see that if $M$ is not orientable, then the torsion subgroup of $H_{n-1}(M, \mathbb{Z})$ is cyclic of order $2$ and $H_n(M, \mathbb{Z}_q)$ is zero if $q$ is odd and is cyclic of order $2$ if $q$ is even? Answer … Read more

Errata in Prof. Rotman AIHA book about projectives in the chain complex category (section 10.5)

EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below. The claim of Prof. Rotman in section 10.5 that split complexes (not necessarily exact) with projective terms are the projectives in the category … Read more

Bott and Tu compact cohomology of the circle “differential forms in Algebraic Topology”

On page 27 of that book, it is claimed that the inclusion map δ which maps a form from the non-empty intersection of two open covers of the circle to the disjoint union of those covers has a one dimensional kernel: δ:H1c(U∩V)→H1c(U⊔V) w=(w1,w2)→(−(jU)∗w,(jV)∗w) where inclusion simply extends the form by zero outside the intersection of … Read more

Origin/Etymology of the term Homology

Does anyone know who was the first person to invent the term homology and what their motivation was? All I could find was that Emmie Noether is accredited with inventing the homology group, and that Poincare made essential contributions to the nascent concept. Although I thought Poincare called topology “analysis situs”, so I doubt he … Read more

Why is homology invariant under deformation retraction/homotopy equivalence?

This is pretty basic, but I can’t find any answers on this site on this already. I want to know why the homology of two spaces X and Y are the same when X deformation retracts/is homotopy equivalent to Y. I would appreciate an intuitive explanation and a formal proof. Specifically I’m working through Hatcher’s … Read more

A question about an application of wedge sum of the reduced cohomology

In this example, I feel confused why we can just need to consider the cohomology ring? Of course, H∗(S2,Z)⨁H∗(S4,Z) is not isomorphic to H∗(CP2,Z). But how is that related to the reduced cohomology ring? I think we still need to prove the reduced case is not isomorphic. Answer This is a typical use of reduced … Read more

Why is the first homology group of the torus is Z⊕Z\mathbb{Z}\oplus\mathbb{Z} instead of Z∗Z\mathbb{Z}*\mathbb{Z} or Z×Z\mathbb{Z}\times\mathbb{Z}?

For the torus below: I would like to compute HΔ1(T). Here is how I did: We have C1=Δ1(T)=Z and C2=Δ2(T)=Z∗Z∗Z am I right? Ci=Δi(T) is always a free abelian group isn’t it? HΔ1(T)=ker∂1im ∂2=Z∗Z∗ZZ=Z∗Z, where ∗ denotes the free product. But the working solution given in Hatcher’s page 106 is: My question is: Is my working … Read more

Minus signs in boundary operators (homology)

Let ab be an edge (1-simplex), and ∂1 be the boundary operator. I have seen both ∂1(ab)=−a+b, and also ∂1(ab)=a−b in various books. My intuition is that both should be equivalent when it comes to defining homology, however I am not very clear why they should be equivalent. Thanks for any enlightenment. Answer We have … Read more

Homology and cohomology of compact vs non compact spaces

What is special about the homology of the compact spaces, what are the most elementary properties that are verified by homology of compact spaces that are not true for the homology of non compact spaces ? I’m asking because i read that Euler characteristic is not easily defined for non compact spaces while it is … Read more