I guess I know those two extensions, I know the classification that you mention, therefore I know that I do not have to worry about others.

$b_n(X) \geq b_n(Y)$. Applying a universal coefficient theorem then rules out the second case, and $H_{n-1}$ must be $\mathbb Z$. 0000003974 00000 n orbifold cohomology    0000003032 00000 n 0000008731 00000 n 0000005868 00000 n If X is a complex of dimension n and Y an «-sphere, then by the Hopf Theorem / is determined up to homotopy by its induced cohomology homomorphism with integer coefficients. 0000021328 00000 n 0000010330 00000 n But how do we compute $H_{n-1}(P^n-p)$? 167 0 obj <> endobj HΓ n (G,A)=Tor Pn(Z,A)and Hn Γ (G,A)=Extn P (Z,A)for n 0, where G and Γ act trivially on Z. An addendum to a paper of Lanteri and Struppa, Can. Denote real projective space by $P^n.$ I want to compute $H_*(P^n - p)$ (for some $p \in P^n$) given that I already know $H_*(P^n).$ Now we know that, as for any $n$-manifold, the local homology groups $H_i(P^n,P^n - p)$ at $p$ are zero except when $i = n$, in which case $H_n(P^n,P^n - p) \cong \mathbb{Z}.$ (This is seen by excising out the complement of an Euclidean ball around $p$.) 0000020470 00000 n This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. 0000008915 00000 n It is clear that HΓ n (G,A)∼=Hn(G,AΓ) and Hn Γ (G,A)∼=Hn(G,AΓ) for n 0, if Γ What is the fundamental group of a hypersurface? However, I'm trying to get more comfortable with explicit computations with exact sequences, so I'm looking for feedback on my ideas above.

0000006161 00000 n


0000005196 00000 n vi CONTENTS 5 APPLICATION: VECTOR FIELDS ON SPHERES 39 k-fieldsand Vn,k+ 1 39 A cell decomposition ofVn,k 40 The cohomology ofPn,k 43 6 THE STEENROD ALGEBRA 45 Graded modules and algebras 45 The Steenrod algebra.si 46 Decomposable elements 47 The diagonal map of.si 47 Hopfalgebras 49 The dual ofthe Steenrod algebra 50 Algebras over a Hopfalgebra 52 The diagonal … 0 Why are end-of-file and end-of-input-signal treated differently by sha256sum? subject classification    Hmmm, the commutative diagrams aren't behaving too well!

THE COHOMOLOGY OF BSOn AND BOn WITH INTEGER COEFFICIENTS EDGAR H. BROWN, JR. ABSTRACT. 0000020099 00000 n Think about the cell decomposition of $\Bbb{RP}^n$; it's obtained from $\Bbb{RP}^{n-1}$ by adding an $n$-cell by the projection $S^{n-1} \to \Bbb{RP}^{n-1}$.

It has already been established that this is not necessary to compute the homology of $P^n\setminus p$, but let's still have a look. 0000003413 00000 n %%EOF 0000003318 00000 n


@Sasha: sure. Using Poincaré duality, from this it follows that all Betti numbers of $X$ and $Y$ are equal, except possibly the middle Betti number, for which we have 0000004068 00000 n

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