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...th manifolds, submanifolds, normal bundles and tubular neighbourhoods: see e.g. \cite{Milnor&Stasheff1974|Ch.11 & Ch 18}. ...es and explain how they fit into the overall development of the seminar: i.e. indicate results from previous talks you build and/or results in future ta

... every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly [[amphicheiral]]. * In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy

* if n > 1 then $\pi_1(X_n(\underline{d})) = \{ e \}$, ** the ring $H^*(X_n(\underline{d}))$ is determined by $n$, $d$ and $e(\underline{d})$,

...r first used a $\Zz_7$-invariant, called the $\lambda$-invariant, to show, e.g. that $\Sigma^7_{1, 2}$ is not diffeomorphic to $S^7$. A little later Ke ...ta_n}{\longrightarrow} \pi_n(G/O) \stackrel{\sigma_n}{\longrightarrow} L_n(e) \to \dots~.$$

...cal'' if it is path connected and all its higher homotopy groups vanish, i.e., $\pi_n(X)$ is trivial for $n \ge 2$. ...g geometric constructions or examples lead to aspherical closed manifolds, e.g., non-positively curved closed manifolds, closed surfaces except $S^2$ an

Let $\xi: E \to X$ be a rank r vector bundle classified by $\xi : X \to BO(r)$. A $B_r If $E$ is a vector bundle, let $T(E)$ denote its [[Wikipedia:Thom_space|Thom space]]. Recall that that a fibre

...ories $E$. Even periodic theories are complex orientable which means that $E^0({\mathbb C}P^\infty)$ carries a formal group. The description of $E_*BStr $$(E\wedge BU\left<6 \right>_+)^0( {\mathbb C}P^\infty \times {\mathbb C}P^\inft

...ms were created by such classical figures as J. W. Alexander, H. Hopf, <!--E. van Kampen, K. Kuratowski, S. MacLane,--> L. S. Pontryagin, R. Thom, H. Wh ... 2 we record some of the dimension ranges where no knotting is possible, i.e. where any two embeddings of $N$ are isotopic.In Section 3, we establish no

...e, and for the corresponding action of the group $E^6_D(S^3)$ on the set $E^6_D(N)$, see e.g. \cite[$\S$4]{Skopenkov2016c}.

...be any integer in $\{2,\ldots,\mathrm{h}(w)\}$. For any $\omega\in\mathcal{E}_m(w)$ and any $x\in H^2(G,2;\,\Zz/2^{m-1})$, and assuming relation ($R_1$) * $\Gamma: \mathcal{E}_m(w) \to \Zz/2^{m-1}$,

... homeomorphic to $S^{2n+1}$; hence glueing the cone of $\partial E'$ onto $E'$ produces a $(2n+1)$-manifold $N$ equipped with a homotopy equivalence to ...ial E\cong S^{2n+1}$ on the boundary. As $\partial E'$ is null-bordant in $E$, the surgery obstruction of $\gamma$ is zero. An additional argument shows

...(\Zz)$ is non-degenerate and unimodular, $N$ is a direct summand in $M$, i.e., $M = N\oplus N^\bot$. Now we restrict $q$ to $N^\bot$ and apply the induc (E) for any pair $(r,8s)$ there is an even unimodular symmetric indefinite for

...mology sphere]] if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$. If $n\ge 3$, the [[Generalized P ... link $K$ is the sphere bundle of the tangent bundle of the $n$-sphere, i.~e.~the [[wikipedia:Stiefel manifold|Stiefel manifold]] $V_{n+1,2}(\mathbb R)$

=== The E₈ manifold === ...-manifold with definite intersection form $q_X$. Then $q_X$ is diagonal, i.e. it is the direct sum of the 1-dimensional forms $( \ 1 \ )$ in the positiv

...e trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space

...nvariant for stable vector bundles $E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the sta

- For statements like Theorem, Lemma, Definition etc., use e.g. - For references, use e.g. {{cite|Milnor1958b}}.

==[E] Talks 12-14==

Förderung der Wissenschaften e.V. :Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V.

:Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. :Hofgartenstraße 8, 80539 München