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be7bdf177ccb417f8360db08b7c70f63e3cb6283 | subsection | 898 | 1,121 | Connective global | The morphisms \Omega \operatorname{eig} and \Omega ^\bullet \tilde{\lambda }_{{\mathbf {ku}}}
are loop maps, so they induce homomorphisms with respect to the
group structure by concatenation of loops. This shows that three of the four
maps in (REF ) are homomorphisms of abelian monoids.
Since the right vertical map is ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f181de32062a61e6190107d6c4c13b1fb4fe22bc | subsection | 899 | 1,121 | Connective global | A conjugation involution of the ultra-commutative ring spectrum {\mathbf {ku}}
was defined
in Construction REF .complex conjugation!on {\mathbf {ku}}We denote by {\mathbf {K}}_G(A){\mathbf {K}}_G(A) - equivariant K-group of Aequivariant K-theory
the G-equivariant K-group of A,
i.e., the group completion (Grothendieck g... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d76bf748269c8aab673f431bb04e5dc141e07540 | subsection | 900 | 1,121 | Connective global | If A has finite isotropy groups,
then the homomorphism [-] :{\mathbf {K}}_G(A)\longrightarrow {\mathbf {ku}}_G^0(A_+) is an isomorphism.(i)
We recall from Example REF
the multiplicative Grassmannian \mathbf {Gr}_\otimes ^{\mathbb {C}}multiplicative Grassmannian
with values\mathbf {Gr}_\otimes ^{\mathbb {C}}(V)\ = \ {\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4ec44ddb443e11d021e601bec820f4ef074e10fb | subsection | 901 | 1,121 | Connective global | The morphism of orthogonal spaces
c :\mathbf {Gr}^{\mathbb {C}}\longrightarrow \Omega ^\bullet {\mathbf {ku}} defined in (REF )
has an extensionc_\otimes \ : \ \mathbf {Gr}_\otimes ^{\mathbb {C}}\ \longrightarrow \ \Omega ^\bullet {\mathbf {ku}}\ ,defined by the same formula as for c, namelyc_\otimes (V) \ : \ \mathbf ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7f1da4dc72409d1e5d66ed4bb624dc4126c30fcd | subsection | 902 | 1,121 | Connective global | The upper left triangle commutes because
the tautological bundle on \mathbf {Gr}^{\mathbb {C}}_\otimes (V)
restricts to the tautological bundle on \mathbf {Gr}^{\mathbb {C}}(V)
along the map i(V):\mathbf {Gr}^{\mathbb {C}}(V)\longrightarrow \mathbf {Gr}^{\mathbb {C}}_\otimes (V).
The left vertical map is bijective
by P... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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-... |
85b5297575c9fe22d1d065897e9c7b381b68863f | subsection | 903 | 1,121 | Connective global | Since this map is a group completion of abelian monoids, already
the homomorphisms [-]\circ {\mathbf {K}}_G(h) and {\mathbf {ku}}_G^0(h_+)\circ [-] agree,
by the universal property of group completions.The compatibility with complex conjugation and restriction along
group homomorphisms follow the same pattern.
The conj... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9550b9882a6bcc2dd6fd5aebf0f52718c10f42a2 | subsection | 904 | 1,121 | Connective global | This bundle is isomorphic to f^\star (\gamma _{\operatorname{Sym}(0)}^{\mathbb {C}}) for
the constant map f:A\longrightarrow G r^{\mathbb {C}}(\operatorname{Sym}(0)) with value {\mathbb {C}},
the constant (and only non-trivial) summand in the symmetric algebra associated
to the 0-dimensional G-representation.
The assoc... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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23c79abb09e1ef3228c119b477782cfc5b835a6e | subsection | 905 | 1,121 | Connective global | The tensor product bundle f^\star (\gamma ^{\mathbb {C}}_V) \otimes g^\star (\gamma ^{\mathbb {C}}_W)
is then classified by the compositeA \ \xrightarrow{} \ G r^{\mathbb {C}}(V_{\mathbb {C}})\times G r^{\mathbb {C}}(W_{\mathbb {C}})\ &\xrightarrow{}\ G r^{\mathbb {C}}(V_{\mathbb {C}}\otimes W_{\mathbb {C}}) \\
&\xrigh... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7b52f498bcfbfe20a1e5574592620540cce254c4 | subsection | 906 | 1,121 | Connective global | The map \langle -\rangle : [A,\mathbf {Gr}^{\mathbb {C}}]^G\longrightarrow {\mathbf {K}}_G(A) is also a group completion
(by Theorem REF , or rather its complex analog),
so the unique extension [-]:{\mathbf {K}}_G(A)\longrightarrow {\mathbf {ku}}^0_G(A_+) is an isomorphism.
equivariant K-theory|)We specialize Theorem ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c429398830901fdfa66315010f10b6e0c2e4fbe8 | subsection | 907 | 1,121 | Connective global | Moreover, the map [-] is an isomorphism whenever the group G is finite.The fact that the map is a ring homomorphism, compatible with restrictions,
compatible with complex conjugation
and an isomorphism for finite groups is a special case of
Theorem REF for a one-point G-space.Now we show that the maps [-] are compatibl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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eefe887e4149da3c52dd2d1c71785ab0efbd1932 | subsection | 908 | 1,121 | Connective global | Then the following diagram commutes:@C=11mm{
\pi _0^H({\mathbf {ku}}) &
\pi _0^H({\mathbf {ku}}\wedge G/H_+)[d]^{\pi _0^H(\alpha )} [l]^-{\pi _0^H({\mathbf {ku}}\wedge l)}&
\pi _0^G({\mathbf {ku}}\wedge G/H_+)[d]^{\pi _0^G(\alpha )}
[l]^-{\operatorname{res}_H^G}[r]^-{\pi _0^G({\mathbf {ku}}\wedge \nabla )}
@<-.4ex>@/_1... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6c3a87f6bb1edd8b9dd2c07dbfd0621c7c7fde54 | subsection | 909 | 1,121 | Connective global | By adjointness, this means that the compositeS^V\ \xrightarrow{}\ {\mathbf {ku}}[G/H](V)\ &= \ {{C}}(\operatorname{Sym}(V_{\mathbb {C}});S^V\wedge (G/H)_+)\\
&\xrightarrow{}\ \operatorname{map}^H(G,{{C}}(\operatorname{Sym}(V_{\mathbb {C}});S^V))\ = \ \operatorname{map}^H(G,{\mathbf {ku}}(V))is G-equivariantly null-homo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bb1b43755e30f57a65037ec2c1aba16e2c44172e | subsection | 910 | 1,121 | Connective global | We define a class[W]_H^G \ \in \ \pi _0^G({\mathbf {ku}}[G/H])by specifying a representative G-mapS^V \ \longrightarrow \ {{C}}(\operatorname{Sym}(V_{\mathbb {C}}), S^V\wedge (G/H)_+)
\ = \ {\mathbf {ku}}[G/H](V)byv\ \longmapsto \ [j(\operatorname{map}^H(H g_1^{-1},W)),\dots ,j(\operatorname{map}^H(H g_m^{-1},W));\, v\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.... |
ca0504ff344e838d455cfff992f2d92c7a8115d0 | subsection | 911 | 1,121 | Connective global | The relation{{C}}(\operatorname{Sym}(V_{\mathbb {C}}), S^V\wedge l_+)
[j(\operatorname{map}^H(H g_k,W));\,v\wedge g_k H&]_{k=1,\dots ,m} \\
&=\ [j(\operatorname{map}^H(H,W));\, v]shows that\pi _0^H({\mathbf {ku}}[l])(\operatorname{res}^G_H([W]_H^G))\ = \ [W] \ .The commutativity of the left part of diagram (REF ) then ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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07395e7f85d18c1d75f50a7358f41bd9695cb333 | subsection | 912 | 1,121 | Connective global | The class [W] is then represented by the G-map[j(W);-]\ : \ S^V\ \longrightarrow \ {\mathbf {ku}}(V)\ .The mapJ \ : \ W^{\otimes m} \quad &\longrightarrow \qquad \operatorname{Sym}(V^m_{\mathbb {C}}) \\
w_1\otimes \cdots \otimes w_m &\longmapsto (j(w_1),0,\dots ,0)\cdot (0,j(w_2),0,\dots ,0)\cdot \ldots \ \cdot (0,\dot... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c79026531a077eefb6a1f9267c55f09410d09962 | subsection | 913 | 1,121 | Connective global | The map is multiplicative because dimension
is multiplicative on tensor products.The infinite symmetric product spectrum includes
by a global equivalence of ultra-commutative ring spectra S\! p^\infty \longrightarrow {\mathcal {H}}{\mathbb {Z}}
into the Eilenberg-Mac Lane spectrum
of the integers,Eilenberg-Mac Lane spe... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0f4c88435cad9b5dfc61e8b2c544575dc0c7f138 | subsection | 914 | 1,121 | Connective global | The compositeS^{u W}\ \xrightarrow{}\ {\mathbf {ku}}(u W)\ \xrightarrow{} \ S\! p^\infty (S^{u W})is the map\delta ^n(u W)\ :\ S^{u W}\longrightarrow S\! p^{\infty }(u W)\ .Hence we conclude that\pi _0^G(\dim )[W] \ = \ [\delta ^n(u W)]\ = \ \delta ^n_*(1)\ = \ n\cdot 1\ = \ \dim (W)\cdot 1\ .This proves the propositio... | {
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"raw": "G. Segal, The representation ring of a compact Lie group. Inst. Hautes Études Sci. Publ. Math. 34 (1968), 113–128.",
"source_ref_id": "93a5d76... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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71a0e3a69863bbcb77918c27858cb901496c1db3 | subsection | 915 | 1,121 | Connective global | The fixed set of such a regular element g on G/N then
consists of a single coset, namely \gamma N. So\chi _{i_!(1)}(g)\ = \ \chi _1(\gamma ^{-1}g\gamma )\ = \ 1 \ .Since the character is continuous and regular elements are dense,
the character of i_!(1) is constant with value 1.
Since the character determines the repre... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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440f1999b477c41f2229a922cab7bacddcffcf40 | subsection | 916 | 1,121 | Connective global | However, in the real situation there is no
eigenspace decomposition, hence no direct analog of
the delooping \operatorname{eig}:{\mathbf {U}}\longrightarrow \Omega ^\bullet (\operatorname{sh}{\mathbf {ku}}) of the morphism c.
So to establish the real analog of Theorem REF
(which is used in the proof of Theorem REF ),
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3ea68a36faa3bfc3aefb0768576ab98eb73aaa47 | subsection | 917 | 1,121 | Connective global | Indeed, the complex global projective space {\mathbf {P}}^{\mathbb {C}}=\mathbf {Gr}_\otimes ^{{\mathbb {C}},[1]},
introduced in (REF ), is a multiplicative model of the
global classifying space of U(1).
A configuration of points labeled by vector spaces
of total dimension 1 has to be concentrated on at most one point.... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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31212e96f5b39b58fa7bca92a2777c06a77f6721 | subsection | 918 | 1,121 | Connective global | An element of infinite order in the kernel of the map
{\mathbf {A}}(U(1),C_2)\longrightarrow \pi _0^{C_2}({\mathbf {ku}}) is\operatorname{tr}_e^{C_2}\circ \operatorname{res}^{U(1)}_e - z^* - \operatorname{res}^{U(1)}_{C_2} \ \in \ {\mathbf {A}}(U(1),C_2) \ ,where z:U(1)\longrightarrow C_2 is the trivial homomorphism.
T... | {
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"s... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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991a28f95378d2c2c6b4ada810f98a8bcb7be6a8 | subsection | 919 | 1,121 | Connective global | As we shall show in Theorem REF
below, the Bott class becomes invertible in the homotopy ring
of the periodic K-theory spectrum {\mathbf {KU}}.\beta - Bott class in \pi _2^e({\mathbf {ku}})We define the Bott class by specifying an explicit representative.
We define a continuous map m:S^{{\mathbb {R}}\oplus {\mathbb {C... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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73569679a1996539835239cf81946aedcc2d7594 | subsection | 920 | 1,121 | Connective global | The standard embedding U(2)\longrightarrow U into the infinite unitary group is 4-connected,
so the composite (REF )
represents a generator of the infinite cyclic group \pi _3({\mathbf {ku}}({\mathbb {R}}),\ast ).
Theorem REF says that {\mathbf {ku}}
is a positive \Omega -spectrum (in the non-equivariant sense),
so in ... | {
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"Stefan Schwede"
] | [
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f8b4bcb0d5193485d0f907279a5d1864be04a852 | subsection | 921 | 1,121 | Connective global | Another way to say this is that
if (e_i)_{i=1,\dots , k} is an orthonormal basis of W, then the vectorse_{i_1}\wedge \dots \wedge e_{i_n}form an orthonormal basis of \Lambda ^*( W)
as the indices run through all tuples with
1\le i_1<\dots < i_n\le k.For w\in W we letd_w\ :\ \Lambda ^*(W)\ \longrightarrow \ \Lambda ^*(W... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0bfc6d18c34a899d90626b02e1538184f5c8b7f5 | subsection | 922 | 1,121 | Connective global | So the adjoint d_w^* is inverse to d_w on the summand
w\wedge \Lambda ^*(W^\perp ), and it vanishes on \Lambda ^*(W^\perp ).
Hence d_w^*\circ d_w is the orthogonal projection onto the summand \Lambda ^*(W^\perp ),
and d_w\circ d_w^* is the orthogonal projection onto the other summand
w\wedge \Lambda ^*(W^\perp ). This ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f4f5637dfd66b494fd790ce770ac4f999bda88a0 | subsection | 923 | 1,121 | Connective global | In this case the elements 1, i\in {\mathbb {C}} form
an orthonormal {\mathbb {R}}-basis of u{\mathbb {C}}; we lete \ = \ 1\otimes [1]\text{\qquad respectively\qquad } f \ = \ 1\otimes [i]be their images in the complexified Clifford algebra {\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(u{\mathbb {C}}).
This Cliff... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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be3def43599dc6f2c161534701b6500d816ac22e | subsection | 924 | 1,121 | Connective global | Indeed, given two hermitian inner product spaces V and W, the following
square commutes:@C=12mm@R=7mm{
{\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(u V\oplus u W) [r]^-{\delta _{V\oplus W}}
[dd]_\cong &
\operatorname{End}_{\mathbb {C}}(\Lambda ^*(V\oplus W)) [d]_\cong \\
& \operatorname{End}_{\mathbb {C}}(\Lamb... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f1b56a82fac8d3a690c1b47a0c0bb6f937c0a781 | subsection | 925 | 1,121 | Connective global | In this description, the Bott class \beta _{G,W} is represented by the triple( D(W)\times \Lambda ^{\operatorname{ev}}(W), D(W)\times \Lambda ^{\operatorname{odd}}(W),\alpha )consisting of the trivial vector bundles over D(W)
with fibers \Lambda ^{\operatorname{ev}}(W) respectively \Lambda ^{\operatorname{odd}}(W),
and... | {
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"source_ref_id"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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459d6268670619079521ce3e375da4b366e1018b | subsection | 926 | 1,121 | Connective global | So by subtracting the class of the trivial vector bundle with fiber
\Lambda ^{\operatorname{odd}}(W) we obtain the equivariant Bott class as a reduced
equivariant {\mathbf {ku}}-cohomology class,\beta _{G,W}\ = \ [ \xi (W) ] - [ S^W\times \Lambda ^{\operatorname{odd}}(W)] \ \in \ {\mathbf {ku}}_G^0(S^W) \ .We invite th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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16a08da9ddf445e8f1a7307a72fab6188a2e814a | subsection | 927 | 1,121 | Periodic global | Our main object of study in this section is periodic global K-theory {\mathbf {KU}},
an ultra-commutative ring spectrum whose G-homotopy type realizes
G-equivariant periodic K-theory, see Construction REF .
The model we use is due to M. Joachim ,
and made of spaces of homomorphisms of {\mathbb {Z}}/2-graded C^\ast -alg... | {
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"raw": "M. Joachim, Higher coherences for equivariant K-theory. Structured ring spectra, 87–114, London Math. Soc. Lecture Note Ser., 315, Cambridge University Press, Cambridge, 2004.",
"source_ref_id... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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80f33a0f5e838fba1543c27b284606160734fb68 | subsection | 928 | 1,121 | Periodic global | We can then decompose A into the \pm 1 eigenspaces of \alpha
and obtain a {\mathbb {Z}}/2-grading of the underlying {\mathbb {C}}-algebra by settingA_{\operatorname{ev}} \ = \ \lbrace a\in A \ | \ \alpha (a)=a\rbrace \text{\quad and \quad }
A_{\operatorname{odd}} \ = \ \lbrace a\in A \ | \ \alpha (a)=-a\rbrace \ .The ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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36062433e5e3545977d6271ed58dba0219c0ab94 | subsection | 929 | 1,121 | Periodic global | So the universal property of s
follows from the fact that C(U(1)) is freely generated,
as an ungraded, unital C^\ast -algebra, by the unitary element z;
indeed, unitary elements are in particular normal and have their spectrum
contained in U(1), so the bijectivity of (REF )
becomes a special case of functional calculus... | {
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"Stefan Schwede"
] | [
"math.AT"
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a158475ae042e1e634cb031bda5cc4dda2cb86e8 | subsection | 930 | 1,121 | Periodic global | This characterization readily implies
the cocommutativity and coassociativity of \Delta .
The cocommutativity of \Delta is with respect to the
graded symmetry automorphism of s\hat{\otimes }s,
which involves a sign whenever two odd elements are interchanged.
An explicit definition of the diagonal morphism \Delta
can b... | {
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... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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08401db0035c5c427fa8943511803be55ac2323c | subsection | 931 | 1,121 | Periodic global | Lemma REF provides the relation(i\cdot \delta _{1\otimes v})\circ (i\cdot \delta _{1\otimes v}) \ = \ - \delta _{1\otimes v}^2\ = \ |v|^2\cdot \operatorname{Id}\ .The universal property of \operatorname{{\mathbb {C}}l}(V) provides a morphism of {\mathbb {Z}}/2-graded {\mathbb {C}}-algebras\operatorname{{\mathbb {C}}l}(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8d7e7ce57207f53199c70c2cb1ffd03e65fda987 | subsection | 932 | 1,121 | Periodic global | The universal property then provides
a unique extension to a morphism of graded {\mathbb {C}}-algebras\mu _{V,W}\ : \ \operatorname{{\mathbb {C}}l}(V\oplus W)\ \cong \ \ \operatorname{{\mathbb {C}}l}(V)\otimes \operatorname{{\mathbb {C}}l}(W)characterized by \mu _{V,W}[v,w] = [v] \otimes 1 + 1\otimes [w]
for all (v,w)\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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83f8dd3e88b0336719fab5b6ee20f8bfc70366f9 | subsection | 933 | 1,121 | Periodic global | Indeed, the {\mathbb {R}}-linear map\psi \ : \ V \ \ \longrightarrow \ \operatorname{{\mathbb {C}}l}(V) \ , \quad v\ \longmapsto \ i\cdot [v]satisfies \psi (v)^2=-|v|^2\cdot 1, so it
extends to a morphism of {\mathbb {C}}-algebras {\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(V)\longrightarrow \operatorname{{\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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c9e0b41759b8b6c254c10b285048bc956ad8f800 | subsection | 934 | 1,121 | Periodic global | The functional calculus maps are multiplicative in the sense that
the following diagram commutes:\begin{aligned}
@C=5mm@R=8mm{
S^V \wedge S^W [dd]_\cong [rr]^-{\operatorname{fc}\wedge \operatorname{fc}} &&
C^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}(V))\wedge C^\ast _{\operatorname{gr}}(s,\operatorname{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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7e4c815a1f962daadb007a73268081398800ef87 | subsection | 935 | 1,121 | Periodic global | Indeed, for the even generating function we have\mu _{V,W}(\operatorname{fc}(v,w)(u_+))\ &= \ \mu _{V,W}( e^{-|v|^2-|w|^2}\cdot 1)\ = \ ( e^{-|v|^2}\cdot 1)\otimes (e^{- |w|^2}\cdot 1)\\
&=\ \operatorname{fc}(v)(u_+)\otimes \operatorname{fc}(w)(u_+)\ = \ (\operatorname{fc}(v)\hat{\otimes }\operatorname{fc}(w))(u_+\otim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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d9a6f4ec90cc7513bcdbf7a34695103b79494ffe | subsection | 936 | 1,121 | Periodic global | We denote by {\mathcal {K}}_V the C^\ast -algebra
of compact operators on the Hilbert space {\mathcal {H}}_V,
see for example .compact operators!on a Hilbert spaceThe orthogonal spectrum {\mathbf {KU}} assigns to a euclidean inner product space V
the space{\mathbf {KU}} - periodic global K-theoryK-theory@K-theory!perio... | {
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"Stefan Schwede"
] | [
"math.AT"
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7da747ed21cea191422dedeac6f5501c258a9e2b | subsection | 937 | 1,121 | Periodic global | This extends to an isometry of the Hilbert space completions{\mathcal {H}}_V\hat{\otimes }{\mathcal {H}}_W \ = \ \widehat{\operatorname{Sym}}(V_{\mathbb {C}})\hat{\otimes }\widehat{\operatorname{Sym}}(W_{\mathbb {C}}) \ &\cong \ (\operatorname{Sym}(V_{\mathbb {C}})\otimes \operatorname{Sym}(W_{\mathbb {C}}))^\wedge \\
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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... |
bf0fb734d19d906a218d5712e9de32d1c1d0ab2f | subsection | 938 | 1,121 | Periodic global | The multiplication map\mu _{V,W}\ : \ {\mathbf {KU}}(V)\wedge {\mathbf {KU}}(W)\ \longrightarrow \ {\mathbf {KU}}(V\oplus W)is now defined as the compositeC^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}(V)\otimes {\mathcal {K}}_V)&\wedge C^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}(W)\otimes ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2450e79b8274c19fe04f2dacc29b87c331e78c8f | subsection | 939 | 1,121 | Periodic global | In other words,\eta _V(v)(f)\ = \ f[v]\otimes p_0\ .The multiplicativity of the unit maps follows from
the multiplicativity (REF )
of the functional calculus maps and the fact that
the isomorphism (REF ) sends p_0\otimes p_0 to p_0.Construction 4.12
The connective and periodic global K-theory spectra are related by a ... | {
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"source_ref_i... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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a58af0953e7878853db5a3cfa6989bd55b90457d | subsection | 940 | 1,121 | Periodic global | These two Hilbert spaces are naturally isomorphic,
as follows. We use the inner product on V to identify it
with its dual space V^\ast , and hence the symmetric algebra
\operatorname{Sym}(V_{\mathbb {C}}) with \operatorname{Sym}(V^\ast _{\mathbb {C}}). Elements of \operatorname{Sym}(V^\ast _{\mathbb {C}})
are complex v... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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34df53e9f8435e0563b83605a0be14c187dbbad9 | subsection | 941 | 1,121 | Periodic global | So all odd-dimensional terms of {\mathbf {KU}} are homeomorphic,
and all even-dimensional terms in positive degrees are homeomorphic.We can also identify the space {\mathbf {KU}}({\mathbb {R}}) as follows. We claim that the mapj({\mathbb {R}})\ : \ {\mathbf {ku}}({\mathbb {R}}) \ \longrightarrow \ {\mathbf {KU}}({\math... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f1b768951f76a1c8694aca170412a8b39b20e224 | subsection | 942 | 1,121 | Periodic global | This implies that the following square commutes:@C=10mm{
{{C}}(\operatorname{Sym}({\mathbb {C}}),S^1)\ = \ {\mathbf {ku}}({\mathbb {R}})[r]^-{j({\mathbb {R}})} [d]_\cong &
{\mathbf {KU}}({\mathbb {R}}) = C^\ast _{\operatorname{gr}}(s, \operatorname{{\mathbb {C}}l}({\mathbb {R}})\otimes {\mathcal {K}}_{\mathbb {R}}) [d]... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3c4ec948afad984cebc2598f90fe24c86d5a5698 | subsection | 943 | 1,121 | Periodic global | Since the Clifford algebra is finite-dimensional, the mapC_0(V)\otimes \operatorname{{\mathbb {C}}l}(V)\ \longrightarrow \ C_0(V, \operatorname{{\mathbb {C}}l}(V)) \ , \quad f\otimes x \ \longmapsto \ \ f(-)\cdot xis an isomorphism of C^\ast -algebras.
Functional calculus (REF )
provides a distinguished graded \ast -ho... | {
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"raw": "N. Higson, E. Guentner, Group C^\\ast -algebras and K-theory. Noncommutative geometry, 137–251, Lecture Notes in Mathematics, Vol. 1831, Springer-Verlag, Berlin, 2004.",
"source_ref_id": "467... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ac9cf7422d25b8a75f298fc1881210bdf50c5cbf | subsection | 944 | 1,121 | Periodic global | We will now establish an analogous property for the C^\ast -algebras
of compact operators on the Hilbert space completions.Construction 4.17 We recall that a {\mathbb {C}}-linear isometric embedding \varphi :{\mathcal {H}}\longrightarrow {\mathcal {H}}^{\prime }
between complex separable Hilbert spaces gives rise to a ... | {
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"arxiv_id": "",
"doi": "",
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"start": 1375
}... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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79ddcd0c98ad3f232d5b651ee17dfe7ecacdb1a7 | subsection | 945 | 1,121 | Periodic global | More precisely, we assume that {\mathcal {H}} and {\mathcal {H}}^{\prime } are
separable complex G-Hilbert space representations.
The C^\ast -algebra {\mathcal {K}}({\mathcal {H}}) then inherits a continuous
G-action by conjugation with the G-action on {\mathcal {H}}.
If \varphi :{\mathcal {H}}\longrightarrow {\mathcal... | {
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"doi": "10.1023/a:1026536332122",
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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df1822012c715acf72d2c03a4be6ca7a0636dcca | subsection | 946 | 1,121 | Periodic global | By the previous paragraph the G-C^\ast -homomorphisms
{\mathcal {K}}(\hat{v})\circ {\mathcal {K}}(\hat{u})={\mathcal {K}}(\widehat{v u})
and {\mathcal {K}}(\hat{u})\circ {\mathcal {K}}(\hat{v})={\mathcal {K}}(\widehat{u v})
are G-homotopic to the respective identity maps.We recall from Definition REF that an orthogonal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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144c8e9d36873826aa96be22434743795b96e9ff | subsection | 947 | 1,121 | Periodic global | So the induced map {\mathcal {K}}(\hat{u}):{\mathcal {K}}_W\longrightarrow {\mathcal {K}}_{V\oplus W}
of compact operators is a G-equivariant homotopy equivalence of C^\ast -algebras
by Proposition REF .The adjoint structure map \tilde{\sigma }_{V,W} factors as the composite:{\mathbf {KU}}(W) \ = \ &C^\ast _{\operatorn... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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db871d2e2fbcefad0a6bd8773e1a3612c4ad7d52 | subsection | 948 | 1,121 | Periodic global | The eigenspace morphism was defined in (REF ).Theorem 4.20
The composite{\mathbf {U}}\ \xrightarrow{} \ \Omega ^\bullet (\operatorname{sh}{\mathbf {ku}}) \ \xrightarrow{} \ \Omega ^\bullet (\operatorname{sh}{\mathbf {KU}})is a global equivalence of orthogonal spaces.As in the proof of Theorem REF
we let \bar{{\mathbf... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9117f5de5155c4e3dd6e23367fa0e973d0c11703 | subsection | 949 | 1,121 | Periodic global | This map factors through the G-mapU(\operatorname{Sym}((V\oplus {\mathbb {R}})_{\mathbb {C}})) \cong {{C}}&(\operatorname{Sym}((V\oplus {\mathbb {R}})_{\mathbb {C}}),S^1) \ \longrightarrow \ \hat{{{C}}}({\mathcal {H}}_{V\oplus {\mathbb {R}}},S^1)\\
&\cong \ C^\ast (s, {\mathcal {K}}_{V\oplus {\mathbb {R}}})\ \cong \ C^... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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af4157635b82f5b6f85e28f323ed0d2cdef93baf | subsection | 950 | 1,121 | Periodic global | This proves the claim because the target is G-equivariantly homeomorphic toC^\ast _{\operatorname{gr}}(s, C_0(V)\otimes \operatorname{{\mathbb {C}}l}(V)\otimes \operatorname{{\mathbb {C}}l}({\mathbb {R}})&\otimes {\mathcal {K}}_{V\oplus {\mathbb {R}}}) \\
_(\ref {eq:Clifford tensor iso}) &\cong \ C^\ast _{\operatorname... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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11298ddda631144954386bffe8d66bccad84c4a9 | subsection | 951 | 1,121 | Periodic global | We recall that {\mathbf {K}}_G(A) denotes the equivariant K-group of a G-space A,
i.e., the Grothendieck group of isomorphism classes of G-vector bundles over A.
A ring homomorphism [-] from this Grothendieck group
to the equivariant cohomology group {\mathbf {ku}}^0_G(A_+)
was defined in (REF ).Corollary 4.22
For eve... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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09d6374dcb93d006fce3d3da78c54cfeab6137ea | subsection | 952 | 1,121 | Periodic global | In the commutative diagram of abelian monoids@C=12mm{
[A,\mathbf {Gr}^{\mathbb {C}}]^G [d]_{[A,\beta ]^G} [r]^-{ [A,c]^G} &
[A,\Omega ^\bullet {\mathbf {ku}}]^G [r]^-{ [A,\Omega ^\bullet j]^G} &
[A, \Omega ^\bullet {\mathbf {KU}}]^G\ =\ {\mathbf {KU}}_G^0(A_+)
[d]^{ [A, \Omega ^\bullet \tilde{\lambda }_{{\mathbf {KU}}}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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53f0d58efa3171bbec52894054ab9172ccb0a2f7 | subsection | 953 | 1,121 | Periodic global | In this case the group {\mathbf {K}}_G(\ast ) becomes
the unitary representation ring \mathbf {RU}(G), and {\mathbf {KU}}_G^0(\ast ) becomes
the 0-th equivariant homotopy group \pi _0^G({\mathbf {KU}}).Theorem 4.23 representation ring!unitary
As G ranges of all compact Lie groups, the composite maps\mathbf {RU}(G)\ \xr... | {
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"Stefan Schwede"
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ef37834942c5f2a1f3c9feea060d7f8b93b54132 | subsection | 954 | 1,121 | Periodic global | Since the maps from \mathbf {RU} to {\underline{\pi }}_0({\mathbf {KU}}) under consideration
do commute with finite index transfers, they commute with the right hand
side of the double coset formula, hence also with the left hand side.
This shows that (REF ) holds after restriction
to every finite abelian subgroup, so ... | {
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"Stefan Schwede"
] | [
"math.AT"
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06d598afb367eeaca178b5ee98d37888aa56fd39 | subsection | 955 | 1,121 | Periodic global | Then by ,
the projection A\times E(cyc\cap G)\longrightarrow A induces an isomorphism{\mathbf {K}}^*_G( A )\ \cong \ {\mathbf {K}}^*_G( A \times E(cyc\cap G) )on equivariant K-groups for every finite G-CW-complex A,
where E(cyc\cap G) is a universal G-space for the family of finite cyclic subgroups
of G.
The Milnor sho... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4438ed46888c40ee09768504922ddf76d142f0b0 | subsection | 956 | 1,121 | Periodic global | So the maps \ \longrightarrow \ \operatorname{{\mathbb {C}}l}(u{\mathbb {C}})\otimes {\mathcal {K}}_{u{\mathbb {C}}} \ , \quad f \ \longmapsto \ f(0)\cdot q\otimes p_0is a {\mathbb {Z}}/2-graded \ast -homomorphism, i.e., an element in the spaceC^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}(u{\mathbb {C}})\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ae20ac06645c6ea415f19ddd963b51d791d2f492 | subsection | 957 | 1,121 | Periodic global | So we can extend the isomorphism to the odd summands by setting\Psi \left(d\otimes \left(\begin{} w & x \\ y & z \end{}\right)\right)
\ = \ i\cdot d e f \cdot \Psi \left(1\otimes \left(\begin{} w & x \\ y & z \end{}\right)\right)\ .The result is an isomorphism of {\mathbb {Z}}/2-graded C^\ast -algebras\Psi \ : \ \opera... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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764378a33ac10045b643e8b7e36d2ecf15341779 | subsection | 958 | 1,121 | Periodic global | So to identify the composite j^{\prime }\circ \operatorname{eig}\circ m we must
calculate these eigenvalues and eigenspaces, and their orthogonal projections.
This is a straightforward exercise in linear algebra:
a direct calculation shows that the matricesp_+ \ = \ \frac{1}{2|v|} \begin{pmatrix} |v|- x & \bar{z} \\ z ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2ebc5f67db34cb435a5eb08aa6cb80466053893b | subsection | 959 | 1,121 | Periodic global | For the even component r_+ the calculation is(\Psi _*\circ j^{\prime }\circ \operatorname{eig}\circ m)(x,z)(r_+)\ &= \ \Psi \big ( r_+[|v|]\otimes p_+ + r_+[-|v|]\otimes p_-\big ) \\
&= \ \Psi \big ( r_+[|v|]\otimes (p_+ + p_-)\big ) \ = \ r_+[|v|]\cdot 1\ = \ r_+[x,z]\ .For the odd component r_- we first observe that\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d7938ab37cadefa39cac299f45f9ffee8be91960 | subsection | 960 | 1,121 | Periodic global | So(\Psi _*\circ j^{\prime }\circ \operatorname{eig}\circ m)(x,z)(r_-)\ &= \ \Psi \big ( r_-[|v|]\otimes p_+ + r_-[-|v|]\otimes p_-\big ) \\
&= \ \Psi \big ( r_-[|v|]\otimes (p_+ - p_-)\big ) \\
&= \ \Psi \left( \frac{2 i [|v|]}{|v|^2+1}\otimes \frac{1}{|v|} \begin{pmatrix}-x &\bar{z}\\ z & x \end{pmatrix}\right) \\
&= ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8828e2594bd84b762c6dc78889c8f18ef3dfc6f7 | subsection | 961 | 1,121 | Periodic global | We claim that the conjugation map \zeta
is homotopic, through graded \ast -homomorphisms, to the identity
of \operatorname{{\mathbb {C}}l}({\mathbb {R}}\oplus u{\mathbb {C}})\otimes M_2.
To show this we define a continuous pathu\ : \ [0,\pi /2] \ \longrightarrow \ \operatorname{{\mathbb {C}}l}({\mathbb {R}}\oplus u{\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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36651c285a636fe13e4e24ecb6dbdda65398f7fa | subsection | 962 | 1,121 | Periodic global | So the multiplication map
-\cdot \lambda : \pi _{k+2}^e({\mathbf {KU}})\longrightarrow \pi _k^e({\mathbf {KU}})
is the effect on homotopy groups of the based continuous map{\mathbf {KU}}(V)\ = \ &C^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}(V)\otimes {\mathcal {K}}_V)\\
&\xrightarrow{} \ C^\ast _{\opera... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f4be195170fd35affa55049f776fbcbae786e7b6 | subsection | 963 | 1,121 | Periodic global | We contemplate the following diagram of continuous based maps:@C=5mm@R=8mm{
S^{{\mathbb {R}}\oplus {\mathbb {C}}} [d]_{\operatorname{eig}\circ m} [rr]^-{\operatorname{fc}} &&
C^\ast _{\operatorname{gr}}(s,\operatorname{{\mathbb {C}}l}({\mathbb {R}}\otimes u{\mathbb {C}}))
[d]^-{\left(-\otimes \left(\begin{}1&0\\0&0\end... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bf412f37fbebae8b38416c04ef83029a6b140e01 | subsection | 964 | 1,121 | Periodic global | By Step 2, the upper left part commutes on the nose,
and by Step 3 the upper right triangle commutes up to based homotopy.
The lower left part commutes because j^{\prime } is the restriction of j({\mathbb {R}}).
The lower right part commutes as well.
So the whole diagram commutes up to based homotopy.
By Step 4, the co... | {
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"doi": "10.1093/qmath/19.1.113",
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"raw": "M. F. Atiyah, Bott periodicity and the index of elliptic operators. Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140.",
"source_ref_id"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0654932e72afb50515ffd8517a3d8f78bb54c633 | subsection | 965 | 1,121 | Periodic global | Joachim defines a morphism of ultra-commutative ring spectra
\alpha :\mathbb {M} \operatorname{Spin}^c\longrightarrow {\mathbf {KU}} from a global equivariant version of the
\operatorname{Spin}^c-Thom spectrum; his morphism refines the Atiyah-Bott-Shapiro orientation,
in the sense that it takes certain tautological Tho... | {
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"doi": "10.1016/j.jpaa.2003.07.008",
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"source_ref_i... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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4e55fb58c6c51c595255889a1b345a83ba7886d7 | subsection | 966 | 1,121 | Periodic global | So more explicitly, we set{\mathbf {ku}^c}\ = \ {\mathbf {KU}}\times _{b({\mathbf {KU}})} b({\mathbf {KU}})^{[0,1]}\times _{b({\mathbf {KU}})} b({\mathbf {ku}}) \ .Since the spectra {\mathbf {KU}}, b({\mathbf {KU}}) and b({\mathbf {ku}})
are ultra-commutative ring spectra and the two morphisms
i_{{\mathbf {KU}}}:{\math... | {
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"source_ref_... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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884c321aff4e85b434604ec4b61d264cc7be348d | subsection | 967 | 1,121 | Periodic global | The natural isomorphisms\pi _{-*}^G(b({\mathbf {ku}})) \ \xrightarrow{} \ {\mathbf {ku}}^*(B G) \text{\quad and\quad }
\pi _{-*}^G(b ({\mathbf {KU}})) \ \xrightarrow{}\ {\mathbf {KU}}^*(B G)of Proposition REF
then show that the map{\underline{\pi }}_*(b j) \ : \ {\underline{\pi }}_*(b({\mathbf {ku}})) \ \longrightarro... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c606b65b2ace5ed81d0557ec0b243e11cc26832d | subsection | 968 | 1,121 | Periodic global | Since the morphism {\mathbf {KU}}\longrightarrow b({\mathbf {KU}}) is multiplicative, the same goes
for the Borel theory b({\mathbf {KU}}). So the morphisms\tilde{\beta }\ : \ {\mathbf {KU}}\wedge S^2 \ \longrightarrow \ {\mathbf {KU}}\text{\quad and\quad }
b(\tilde{\beta })\ : \ b({\mathbf {KU}})\wedge S^2 \ \longrigh... | {
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"source_ref_... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e4d0e68869466eb025f1bd79dbd593c389f99273 | subsection | 969 | 1,121 | Periodic global | In particular, we conclude that{\underline{\pi }}_k({\mathbf {ku}^c}) \ \cong \ {\left\lbrace \begin{array}{ll}
\mathbf {RU}& \text{\ for $k\ge 0$ and $k$ even, and}\\
\, 0 & \text{\ for $k\ge -1$ and $k$ odd.}
\end{array}\right.}More precisely, for every m\ge 0, the composite\mathbf {RU}\ \xrightarrow{}\ {\underline{\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1589d2d6d0c17b20c1b5f0ce6e3a76b9a2a213fb | subsection | 970 | 1,121 | Periodic global | So we conclude that {\underline{\pi }}_{-3}({\mathbf {ku}^c}) = 0.This method can be pushed a little further to also determine
the global functors \pi _{-4}({\mathbf {ku}^c}) and \pi _{-5}({\mathbf {ku}^c});
we refer to for the argument.
The result is that{\underline{\pi }}_{-4}({\mathbf {ku}^c}) \ \cong \ \mathbf {I S... | {
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"Stefan Schwede"
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"math.AT"
] | 2,018 | en | Mathematics | [
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104ed84d0a6027e02799e6baeecbd9628aa5bf63 | subsection | 971 | 1,121 | Compactly generated spaces | In this appendix we recall some background material about compactly
generated spaces, our basic category to work in.
Compactly generated spaces are in particular `k-spaces',
a notion that seems to go back to Kelley's book .
Compactly generated spaces were
popularized by Steenrod in his paper
as a `convenient category ... | {
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{
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"doi": "10.1090/s0002-9947-1937-1501905-7",
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"source_ref_id": "6... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7be78d25e096cc301a12a289d1d679b58f82b0dc | subsection | 972 | 1,121 | Compactly generated spaces | Thus k-spaces can equivalently be defined by the property
that all compactly open subsets are open.We denote by \mathbf {Spc}\mathbf {Spc} - category of topological spaces
the category of topological spaces and
continuous maps. We denote by {\mathbf {K}}{\mathbf {K}} - category of k-spaces
respectively {\mathbf {T}}{\m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0002-9947-1969-0251719-4",
"end": 890,
"openalex_id": "https://openalex.org/W2010050433",
"raw": "M. C. McCord, Classifying spaces and infinite symmetric products. Trans. Amer. Math. Soc. 146 (1969), 273–298.",
"source_ref_id... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5277315c0ed59886e7eba99a3a953043cb3305c9 | subsection | 973 | 1,121 | Compactly generated spaces | Every closed subset of a k-space is a k-space in the subspace topology.
Every locally compact Hausdorff space, and hence every compact space,
is a k-space.
Every first countable space is a k-space.
Every metric space is first countable, and hence a k-space.
If X is a k-space and Y a locally compact Hausdorff space,... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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159d93a4835165ab10aabd29852688c84b5603ab | subsection | 974 | 1,121 | Compactly generated spaces | We let z\in \bar{A} be a point in the closure of A.
The point z has a countable basis of open neighborhoods \lbrace U_n\rbrace _{n\ge 1},
which we can moreover take to be nested, i.e.,U_1 \ \supset \ U_2 \ \supset \ \cdots \ \supset \ U_n \ \supset \ \cdots \ .If the intersection of U_n and A were empty, then
\bar{A}\s... | {
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"Stefan Schwede"
] | [
"math.AT"
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eec24339bd4bab6a51f9d790f36ec199287cbb9f | subsection | 975 | 1,121 | Compactly generated spaces | Hence (f\times K)^{-1}((X\times K)\cap A) is compact in the subspace topology
inherited from C\times K.
Sincef^{-1}(B) \ = \ \lbrace c \in C \ | \ (\lbrace f(c)\rbrace \times K)\cap A \ne \emptyset \rbraceis the projection of (f\times K)^{-1}((X\times K)\cap A) onto C,
the set f^{-1}(B) is closed in C.
Altogether this ... | {
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"Stefan Schwede"
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] | 2,018 | en | Mathematics | [
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bad6b681d199d1a37b1e1166cfd3b4d48607f1b4 | subsection | 976 | 1,121 | Compactly generated spaces | We denote by X\times Y=k(X\times _0 Y) the Kelleyfication
of the product topology; if X and Y are k-spaces, then X\times Y
is a categorical product in the category \mathbf {K}.
Proposition REF (vi)
shows that Kelleyfication is unnecessary if one
of the factors is locally compact Hausdorff.An important example where pr... | {
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"doi": "10.1017/cbo9780511983948",
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"raw": "R. Fritsch, R. Piccinini, Cellular structures in topology. Cambridge Studies in Advanced Mathematics, 19. Cambridge University Press, Cambridg... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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db5c85d87172fe9e638a6eab66a6ea9b635f549f | subsection | 977 | 1,121 | Compactly generated spaces | We letf \ = \ (f_1,f_2) \ : \ K\ \longrightarrow \ Y\times Zbe a continuous map from a compact space.
Then(p\times K)^{-1}((Y\times f_2)^{-1}(A))\ = \ (X\times f_2)^{-1}((p\times Z)^{-1}(A))is closed in X\times K.
The map p\times K:X\times K\longrightarrow Y\times K
is a proclusion by the special case, so
the set (Y\ti... | {
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"doi": "10.1090/s0002-9947-1969-0251719-4",
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"source_ref_i... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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18f766c916d0274c28d7be32386ef652615c955d | subsection | 978 | 1,121 | Compactly generated spaces | The sets \lbrace x\rbrace
and \lbrace y\rbrace are closed in X by part (iii), so
f^{-1}(x) and f^{-1}(y) are disjoint closed subsets of K.
Since compact spaces are normal, there are disjoint open subsets U and V
of K with f^{-1}(x)\subset U and f^{-1}(y)\subset V.
Since X is weak Hausdorff, the sets f(K-U) and f(K-V)
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4f0620f8b1e0e8d9bc4a5cf69c999a777111f952 | subsection | 979 | 1,121 | Compactly generated spaces | Every locally compact Hausdorff space, and hence every compact space,
is compactly generated.
Every metric space is compactly generated.
Every disjoint union of compactly generated spaces is compactly generated.
If X is compactly generated and Y locally compact Hausdorff, then
X\times _0 Y is compactly generated in ... | {
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"source_ref_id... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1df8d75f67425e6223a8fa73778953112134bbb5 | subsection | 980 | 1,121 | Compactly generated spaces | Since X and X/E are k-spaces, the mapp\times p \ : \ X\times X \ \longrightarrow \ (X/E)\times (X/E)is a proclusion by two applications of Proposition REF .
Since E is closed by hypothesis, the relation (REF )
shows that \Delta _{X/E} is closed in (X/E)\times (X/E).
So X/E is weak Hausdorff by Proposition REF .Corollar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a3bf41bcab5de952a91fcd7f1b94258fc10c9021 | subsection | 981 | 1,121 | Compactly generated spaces | In this situation the quotient map X\longrightarrow X/E_{\min }=w(X)
is a homeomorphism.It follows formally from the existence of a left adjoint
to the inclusion \mathbf {K}\subset {\mathbf {T}} that the category of compactly generated spaces
has small limits and colimits;
limits can be calculated in the category \math... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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29e677c111a8bf3cf0fdef59f6ef07946f41d7cb | subsection | 982 | 1,121 | Compactly generated spaces | However, closed subsets of k-spaces
are again k-spaces in the usual subspace topology,
so there is no such ambiguity with the notion of `closed embedding'.Proposition 4.44
Let i:A\longrightarrow X and j:B\longrightarrow Y be closed embeddings between topological spaces.
Then the product maps i\times _0 j:A\times _0 B\... | {
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{
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"doi": "",
"end": 1458,
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"raw": "L. G. Lewis, Jr., The stable category and generalized Thom spectra. Ph.D. thesis, University of Chicago, 1978.",
"source_ref_id": "2e56d470deef8729f559920366ac... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c91a868ca9387f2deb55c6d79314bb8d2c394f02 | subsection | 983 | 1,121 | Compactly generated spaces | If moreover Y and Z are compactly generated, then
so is P, and hence the square is a pushout in {\mathbf {T}}.We adapt the argument given in .
The map j is injective because i is. Indeed, we can choose
a set-theoretic retraction r:Y\longrightarrow X to i (not necessarily continuous),
and then (f r)\cup \operatorname{Id... | {
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"source_ref_id... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.02800... |
0509c6ae9edd0519e84a3a07f43fcf1ce4357f6d | subsection | 984 | 1,121 | Compactly generated spaces | We consider the diagram\lbrace -1,1\rbrace \ \xleftarrow{}\ [-1,0) \cup (0,1] \ \xrightarrow{} \ [-1,1]where all three spaces have the subspace topology of {\mathbb {R}}, and the left map
takes [-1,0) to -1 and it takes (0,1] to 1.
All three spaces are compactly generated, and the pushout P
in the categories \mathbf {S... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3cd8dc11a3d41bcc48984d8c1d3e4b8f9c270062 | subsection | 985 | 1,121 | Compactly generated spaces | Hence\kappa _j^{-1}(\kappa _i(A))\ =\ F(j,k)^{-1}(\kappa _k^{-1}(\kappa _k(F(i,k)(A))))\ =\ F(j,k)^{-1}(F(i,k)(A)) \ .Since F(i,k) is a closed map, this shows that
\kappa _j^{-1}(\kappa _i(A)) is closed in F(j) for all j\in P.
The map \coprod _{j\in P}F(j)\longrightarrow F_\infty
given by \kappa _j on F(j) is a proclu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
e2ced6b7913d6a3e3473af4758776dd2a8daabbe | subsection | 986 | 1,121 | Compactly generated spaces | Every lattice is in particular a filtered poset.Proposition 4.48
Let (P,\le ) be a lattice with the following property:
for every element q\in P and every countable chainp_1\ \le \ p_2\ \le \ \dots \ \le \ p_n\ \le \ \dotsin P with p_n\le q for all n\ge 1,
the sequence is eventually constant.Let F:P\longrightarrow {\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
0e063966d553cbe983dcf16708fd6776482316a2 | subsection | 987 | 1,121 | Compactly generated spaces | Indeed, we start with any choice x_1\in F(p_1) and continue inductively:
since K is not contained in F(p_n), there is an element x_{n+1}\in K\backslash F(p_n).
We must have x_{n+1}\in F(q) for some q\in P, and then p_{n+1}=p_n\vee q
can serve for the inductive step.We set C = \lbrace x_1,x_2,x_3,\dots \rbrace , a count... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
891868052ef43dac9ec761c91cfcd01561648f17 | subsection | 988 | 1,121 | Compactly generated spaces | Then\kappa ^{\prime }_i\circ \psi (i)\circ \alpha ^{\prime }\ =\ \psi _\infty \circ \kappa _i\circ \alpha ^{\prime }\ =\ \psi _\infty \circ \alpha \ =\ \beta |_{\partial D^k}\ = \ \kappa ^{\prime }_i\circ \beta ^{\prime }|_{\partial D^k}\ .Since \kappa ^{\prime }_i is injective this shows that \psi (i)\circ \alpha ^{\p... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5d76046673dc918f9168d68d5465471d97c95f0f | subsection | 989 | 1,121 | Compactly generated spaces | Indeed, any two elements p,q\in P are comparable,
and we suppose that p\le q, the other case being analogous.
Then p\wedge q=p and p\vee q=q, and
the two vertical maps in the commutative square of condition (b) are
identity maps. Hence the square is a pullback.The most familiar special case of Proposition REF
is the p... | {
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{
"arxiv_id": "",
"doi": "10.1090/surv/099",
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"raw": "M. Hovey, Model categories. Mathematical Surveys and Monographs, 63. Amer. Math. Soc., Providence, RI, 1999, xii+209 pp.",
"source_ref_id": "294... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ebe992d7ef2ca47e5c1e995ed4029da927aa39c2 | subsection | 990 | 1,121 | Compactly generated spaces | We consider the functor F:P\longrightarrow {\mathbf {T}} sending a finite subset J\subset I
to the finite wedge \bigvee _{j\in J}X_j.
For J^{\prime }\subset J\subset I, the map \bigvee _{j\in J^{\prime }}X_j\longrightarrow \bigvee _{j\in J}X_j
is a closed embedding
(by direct inspection, or by Proposition REF ),
and pr... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01187411",
"end": 2051,
"openalex_id": "https://openalex.org/W2027107802",
"raw": "D. Puppe, Homotopiemengen und ihre induzierten Abbildungen. I. Math. Z. 69 (1958), 299–344.",
"source_ref_id": "ccc47ed1bb6c23447646e06a0ba15... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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38108bb29a7572dfbccd0fb64daeb0cc763343b1 | subsection | 991 | 1,121 | Compactly generated spaces | The maps
X\wedge Y \ \longrightarrow \ Y\wedge X\ , \quad x\wedge y\ \longmapsto \ y\wedge x
and
(X\wedge Y)\wedge Z \ \longrightarrow \ X\wedge (Y\wedge Z)\ , \quad (x\wedge y)\wedge z\ \longmapsto \ x\wedge (y\wedge z)
are homeomorphisms.(i) In weak Hausdorff spaces all points are closed
(Proposition REF (iii)), ... | {
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{
"arxiv_id": "",
"doi": "10.1093/qmath/15.1.238",
"end": 2179,
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"raw": "R. Brown, Function spaces and product topologies. Quart. J. Math. Oxford Ser. (2) 15 (1964), 238–250.",
"source_ref_id": "14fd6b9a594fd1a... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e2f7ea918f416963f4474142e00c76d20e04163d | subsection | 992 | 1,121 | Compactly generated spaces | Internal function spaces in {\mathbf {K}} and {\mathbf {T}} are given by the set of all continuous maps
endowed with the Kelleyfication
of a slight modification of the compact-open topology.
For weak Hausdorff spaces, the `modified' compact-open topology actually coincides with
the compact-open topology.Construction 4.... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.026631897315382957,
0.01... |
db1012671fb2cc8efea3db3da76e344533630447 | subsection | 993 | 1,121 | Compactly generated spaces | So in this case N(h,U)=W(h(K),U), and the two subbases coincide.We recall that a category
is cartesian closedcartesian closed category
if it has finite products and product with a fix object is a left adjoint.
The proof of the following theorem can be found
in .Theorem 4.55
For all k-spaces X,Y and Z, the natural map\... | {
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"source_ref_id": "2e56d470deef8729f559920366aca... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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ff60cdcbd4564afe79279f729dfad3814a3a3d69 | subsection | 994 | 1,121 | Compactly generated spaces | Since X is a k-space, the map \eta _X is also continuous
with respect to the Kelleyfied topology on the target,
i.e., when considered as a map to \operatorname{map}(Y,X\times Y).We denote the evaluation map by\epsilon _Z \ : \ \operatorname{map}(Y,Z)\times Y\ = \ k( C(Y,Z)\times _0 Y) \ \longrightarrow \ Z\ , \quad \ep... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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6e23c46d7c8e10be3022c1b3a74ed4e7789277fd | subsection | 995 | 1,121 | Compactly generated spaces | The map \epsilon _Z\circ (g,h) coincides with the compositeK \ \xrightarrow{} \ C(Y,Z)\times _0 K \ \xrightarrow{} \ C(Y,Z)\times _0 Y \ \xrightarrow{}\ Z \ .The composite of the last two maps is continuous by the previous paragraph,
so the whole composite \epsilon _Z\circ (g,h) is continuous.At this point we know that... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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e8e4ea8d8a5c73ed124827f0d5aa09003d30541e | subsection | 996 | 1,121 | Compactly generated spaces | Indeed, for an open set U of Y we have
\operatorname{ev}_x^{-1}(U)=N(\text{incl}:\lbrace x\rbrace \longrightarrow X,U)
which is open because a one-point space is compact.The evaluation mapC(X,Y)\ \longrightarrow \ {\prod }^0_{x\in X}\ Y \ , \quad f \ \longmapsto \ \lbrace f(x)\rbrace _{x\in X}is injective, and it is co... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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0958820491c882c1217f09a2a95afaa7d31954a9 | subsection | 997 | 1,121 | Compactly generated spaces | The compositeX \ \xrightarrow{}\ \operatorname{map}(Y,X\times Y) \ \xrightarrow{}
\operatorname{map}(Y,X\wedge Y)takes values in the subspace \operatorname{map}_*(Y,X\wedge Y), so it restricts to a continuous map\eta ^{\prime }_X \ : \ X\ \longrightarrow \ \operatorname{map}_*(Y,X\wedge Y)\ ,which is moreover based.
Th... | {
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"raw": "A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp.",
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"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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