# Bilingual Language Modeling, A transfer learning technique for Roman Urdu

Usama Khalid
AIM Lab, NUCES (FAST)
Islamabad, Pakistan
<EMAIL_ADDRESS>
Mirza Omer Beg
AIM Lab, NUCES (FAST)
Islamabad, Pakistan
<EMAIL_ADDRESS>
Muhammad Umair Arshad
AIM Lab, NUCES (FAST)
Islamabad, Pakistan
<EMAIL_ADDRESS>

###### Abstract

Pretrained language models are now of widespread use in Natural Language
Processing. Despite their success, applying them to Low Resource languages is
still a huge challenge. Although Multilingual models hold great promise,
applying them to specific low-resource languages e.g. Roman Urdu can be
excessive. In this paper, we show how the code-switching property of languages
may be used to perform cross-lingual transfer learning from a corresponding
high resource language. We also show how this transfer learning technique
termed Bilingual Language Modeling can be used to produce better performing
models for Roman Urdu. To enable training and experimentation, we also present
a collection of novel corpora for Roman Urdu extracted from various sources
and social networking sites, e.g. Twitter. We train Monolingual, Multilingual,
and Bilingual models of Roman Urdu - the proposed bilingual model achieves 23%
accuracy compared to the 2% and 11% of the monolingual and multilingual models
respectively in the Masked Language Modeling (MLM) task.

##  1 Introduction

Corpus | Reference | Sentence Count | Word Count
---|---|---|---
Urdu NER | Khana et al. (2016) | 1,738 | 49,021
COUNTER | Sharjeel et al. (2017) | 3,587 | 105,124
Urdu Fake News | Amjad et al. (2020) | 5,722 | 312,427
Urdu IMDB Reviews | Azam et al. (2020) | 608,693 | 14,474,912
Roman Urdu sentences | Sharf and Rahman (2018) | 20,040 | 267,779
Roman Urdu Twitter | Proposed | 3,040,153 | 54,622,490
Table 1:  Statistics of the collected Urdu and Roman Urdu corpora. The Urdu
corpora have all been cleaned and transliterated to Roman Urdu. In addition to
this a novel corpus for Roman Urdu has also been proposed.

Most ground breaking research nowadays focuses on Multilingual language
modelling Conneau and Lample (2019); Conneau et al. (2020); Pires et al.
(2019); Khawaja et al. (2018); Beg (2008); Beg and Dahlin ; Farooq et al.
(2019b). This is also desirable from an industrial perspective where most app
makers would like to easily scale to a global audiences. However a lot of
research is also being done on pretraining large monolingual language models
for each language

Virtanen et al. (2019); de Vries et al. (2019); Baly et al. (2020); Yu and
Arkhipov (2019); Polignano et al. (2019); Canete et al. (2020); Martin et al.
(2020); Le et al. (2020); Rani et al. (2015); Beg (2009, 2007); Koleilat et
al. (2006). This can be desirable for achieving better and faster performance
as compared to cross and multilingual language modeling. It has also been
shown by recent works that monolingual models tend to outperform their
multilingual counterparts of the same size in similar settings de Vries et al.
(2019); Martin et al. (2020); Virtanen et al. (2019); Pyysalo et al. (2020);
Farooq et al. (2019a); Zafar et al. (2019a, 2018); Thaver and Beg (2016).
Multilingual is also not always desirable, like when interested only in
improving performance for a specific language. This is because Multilingual
models suffer from the capacity dilution problem Arivazhagan et al. (2019)
also termed as the curse of multilinguality Conneau et al. (2020). This is can
be handled by increasing model capacity but at the cost of memory and
inference time. As the number of languages increase this leaves lesser space
for each language.

Roman Urdu is a code-switched language Beg et al. (2006) formed by a mixture
of Urdu and English. It shares the vocabulary and uses the same Latin script
as English and other words are transliterated from Urdu Alvi et al. (2017);
Zafar et al. (2020). It is widely used on many social media platforms and chat
apps mainly in South Asian countries like India and Pakistan. However it is a
resource starved language as there is not a single corpus, tools or techniques
to create Large Pretrained Language models and enable out-of-the-box Natural
Language Processing (NLP) tasks Beg and Dahlin . In this research we take a
bilingual language modeling approach instead of a multilingual one for Roman
Urdu and show that its code-switching property with English can be
intelligently used to obtain a better performance as compared to Mono and
Multi lingual models. The main contributions of this work are as follows:

  1. 1.

We show how performance of a specific low-resource language can be improved by
taking advantage of its code-switching property and a typo-logically similar
high-resource language.

  2. 2.

We introduce a new method for producing Bilingual models from Monolingual
models using additional pretraining.

  3. 3.

We apply the proposed Bilingual Language modeling techniques to Roman Urdu and
show that significant performance gains can be achieved as compared to
Monolingual modeling.

  4. 4.

We also show how languages can be added to the linguistic space of Mono and
Multilingual models by using vocabulary augmentation and additional
pretraining.

  5. 5.

We apply the proposed Bilingual Language modeling techniques to Roman Urdu and
show that significant performance gains can be achieved as compared to
Monolingual modeling

  6. 6.

We propose a novel collection of Roman Urdu corpora gathered from various
sources and also transliterated from Urdu.

  7. 7.

We make the corpora and pretrained Mono, Bi and Multi lingual BERT and RoBERTa
models publicly available.

The paper is organized as follows. In section 2 we explain in detail the
pretraining methodology and dataset used. Section 3 discusses the evaluation
methodology. Finally the Related work is discussed in section 4.

Figure 1: An example of vocabulary augmentation. Notice how the words driver
and show retain positions in augmented vocabulary from English.

##  2 Bilingual Language Modeling

###  2.1 Vocabulary Augmentation

The main contribution of this paper which enables the cross-lingual transfer
learning Baig et al.  involves vocabulary augmentation. This involves changing
the existing vocabulary of a pretrained model using a specific criteria. After
the vocabulary augmentation additional cycles of pretraining are carried out
to produce the final model.

The vocabulary augmentation technique Seth and Beg (2006) can in theory be
applied to any low resource language which shares vocabulary with a high
resource language Awan and Beg (2021). In this research we apply this
technique to Roman Urdu, a language which shares a significant amount of
vocabulary Javed et al. (2019) with English and show that we achieve
significantly improved performance. Roman Urdu uses the same Latin script as
English and many other Languages in addition to this Roman Urdu also consists
of many English words such that these are often used interchangeably Beg and
Beek (2013) in a code switched mode Bangash et al. (2017). This opens up
exciting opportunities when building a Roman Urdu Language model. An abstract
representation of this idea is also shown in Figure 2. Consider the sentence
Abbas school main parhata hai (Abbas teaches in a school). In this sentence
the English model knows the context of school. Thus is able to learn better
word representations for the surrounding words as compared to training the
model from scratch Naeem et al. (2020).

The advantage of this cross-lingual transfer Sahar et al. (2019) is that it
will enable a significantly improved performance for Roman Urdu without the
need of extensive data and training cycles. An example of augmenting a small
set of Roman Urdu vocabulary with English Qamar et al.  is shown in Fig. 1.
The words driver and show are common in both languages thus their positions
from the original language (English) are retained.

The vocabulary augmentation method Uzair et al. (2019) can be extended to any
language which satisfy certain conditions. The augmentation method and its are
described in detail as follows. Consider two languages $x$ and $y$ where $x$
is a low resource language and $y$ is high resource. Vocabulary augmentation
can be applied for $x$ if equations 1 and 2 hold true. $L(x)$ is the super set
of all the words of language $x$ and produces the vocabulary vector
$V(x)=[x_{1},x_{2},x_{3},...,x_{n}]$. Consider the vocabulary vector
$V(y)=[y_{1},y_{2},y_{3},...,y_{n}]$ given $L(x)\cap L(y)=\\{y_{1},y_{3}\\}$,
the augmented vocabulary vector $V(z)$ will be produced as follows
$V(z)=[y_{1},x_{2},y_{3},x_{4},...,x_{n}]$. The positions of the common
elements between $V(x)$ and $V(y)$ will not be changed while the other
elements from $V(x)$ will be added without changing their position to $V(z)$.

$L(x)\cap L(y)\neq\emptyset$ (1)

$L(y)\subset L(x)$ (2)

###  2.2 Training and Architecture

This section explains the steps to transform data Zafar et al. (2019b) for
pretraining and the changes made to the pretraining cycles to enable cross-
lingual transfer from English to Urdu. We also discuss the architectures used
to perform pretraining for Mono, Bi and Multilingual models.

Architecture. The architectures used in this research are mainly based on BERT
and RoBERTa. All types of pretraining cycles use the BASE architecture Beg and
Van Beek (2010) which consists of 12 layers, 768 hidden nodes, 12 attention
heads and 110M parameters. The pretraining tasks of MLM (Masked Language
Modeling) Dilawar et al. (2018), NSP (Next Sentence Prediction) Javed et al.
(2020b) and SOP (Sentence Order Prediction) use uncased vocabulary. These
Uncased models typically have better performance overall. However cased
versions are useful Asad et al. (2020) for tasks such as Part-of-Speech
tagging or named entity recognition where the case of a letter encodes useful
information.

Figure 2: An abstract representation of mapping new Roman Urdu sentences to an
existing English representation space.

Pretraining. Multilingual models Karsten et al. (2007) such as XLM and XLM-R
have been trained on hundreds of languages of which Roman Urdu is not a part
of. One way to solve this is using cross-lingual transfer which involves fine-
tuning a Multilingual model on a specific Roman Urdu task or zero-shot cross-
lingual transfer Beg et al. (2019) where model is fine-tuned on an English
task and tested on Roman Urdu tasks. However this type of cross-lingual
transfer is bound to yield a higher perplexity as the model has seen very less
examples of Roman Urdu and the representation space is shared with many other
languages.

To overcome both these issues we perform cross-lingual transfer Javed et al.
(2020a) during pretraining phase using vocabulary augmentation and cut down
the languages to a minimal. So the model’s representation space only contains
the high resource language the transfer is being made from and the low
resource language to which the transfer is being made. To enable the cross-
lingual transfer during training phase we run additional pretraining cycles.
This involves running some steps of the pretraining phase for the new language
after augmenting Roman Urdu vocabulary with English.

This notion of cross-lingual transfer Beg (2006) can also be applied to
Multilingual models however the learning space is shared with many other
languages and the curse of multilinguality comes into play. These hypothesis
are confirmed by the three types of pretraining Experiments we perform. The
first type involves training from scratch. The second type involves additional
pretraining of a Monolingual English model and the third type involves
additional pretraining of a multilingual model. The results of these
experiments are discussed in detail in Section 3.

###  2.3 Pretraining Dataset

This section explains the data gathering process and the preprocessing steps
applied to transform the data in a suitable form as expected by the Byte Pair
Encoder (BPE).

Collection. Roman Urdu as mentioned earlier is a resource starved language,
little Zahid et al. (2020); Majeed et al. (2020); Khawaja et al. (2018) or no
work has been done for Roman Urdu and almost no large publicly available
dataset exists. Therefore to enable pretraining of Large Transformer based
language models we propose a novel collection of corpora cleaned and
transliterated to Roman Urdu Tariq et al. (2019). In addition to this we also
propose a novel Roman Urdu dataset consisting of 3 million Roman Urdu
sentences and 54 million tokens. This dataset has been scraped from twitter
where tweets contain at-least a small amount of Roman Urdu. The statistics of
the Twitter scraped dataset and all other datasets is shown in Table 1. All
the above mentioned datasets have also been made publicly available.

Transliteration. Most of the collected datasets are taken from prominent
research works on Urdu while a few were taken from research works on Roman
Urdu Sharf and Rahman (2018); Arshad et al. (2019). The collected Urdu
datasets Khana et al. (2016); Sharjeel et al. (2017); Amjad et al. (2020);
Azam et al. (2020); Nacem et al. (2020) have then been transliterated to Roman
Urdu. The transliteration process involves changing the Urdu Devanagri script
to the Latin script used by Roman Urdu. All of the transliteration is
performed using ijunoon’s Urdu to Roman Urdu transliteration API
111https://www.ijunoon.com/transliteration/urdu-to-roman/.

##  3 Experiments

In this section we describe the methods used for evaluation and discuss the
results obtained for the three different types of pretraining.

Model | MLM | NSP
---|---|---
BERTMonolingual | 0.02 | 0.53
BERTMultilingual | 0.11 | 0.91
BERTBilingual | 0.23 | 0.95
Table 2:  A comparison of MLM and NSP accuracies for three types of BERT
models. As proposed the Bilingual model achieves the highest performance.

###  3.1 Evaluation

Large Pretrained language models are usually evaluated on a range of NLP tasks
by fine tuning the model on each of the specific task. Previously BERT and
BERT like models have been evaluated on the General Language Understanding
Evaluation (GLUE) benchmark Wang et al. (2018) which consists of a range of
Natural Language Understanding (NLU) tasks. The tasks range from simple
semantic analysis, textual entailment to complex reading comprehension tasks
such as in the SQuAD Rajpurkar et al. (2016, 2018) and QNLI Wang et al. (2018)
tasks. Other stricter evaluation tasks have also been proposed over time which
consist of the SQuAD v2.0 Rajpurkar et al. (2018) and the SWAG Zellers et al.
(2018) datasets.

As research progressed, larger and better models like GPT Radford et al.
(2018, 2019), XLNET Yang et al. (2019) achieved close to human level
performance on the GLUE benchmark. This lead to the introduction of SuperGLUE
Wang et al. (2019) a set of more challenging tasks for producing better
language models. However these evaluation tasks were only for English models
and Multilingual or Monolingual models in other languages could not be
evaluated. As the trend towards multilingual and universal language modeling
increased this lead to the creation of Multi-task and Multi-lingual benchmarks
like Xtreme Hu et al. (2020) and XGLUE Liang et al. (2020). Despite of the
abundance of Multilingual and Multitask evaluation sets covering hundreds of
languages no dataset provides support for Roman Urdu. Thus for evaluation of
the performance of our pretrained models we use the validation metrics
calculated during the pretraining phase of each model. The validation dataset
is based on data that is put aside during the creation of pretraining data.

The Evaluation metrics include two tasks for BERT based models namely Masked
Language Modeling (MLM) and Next Sentence Prediction (NSP) while the RoBERTa
models are evaluated in terms of MLM loss and MLM perplexity. The Masked
Language modeling task involves predicting a randomly masked word. A word is
masked randomly about 15% of the time for each given sentence. The Next
Sentence prediction task involves predicting whether two given sentences are
consecutive in the correct order i.e. sentence 2 comes after sentence 1. The
MLM perplexity is calculated using eqn. 3. This represents how sure the model
is when predicting the masked word.

$PP(W)=\frac{1}{P(w_{1},w_{2},...,w_{N})^{\frac{1}{n}}}$ (3)

Model | Train loss | Valid loss | Train perplexity | Valid perplexity
---|---|---|---|---
RoBERTaMonolingual | 2.01 | 2.46 | 4.08 | 5.49
RoBERTaBilingual | 1.72 | 2.19 | 3.24 | 4.56
Table 3:  Loss and perplexity for the MLM pretraining task. The proposed
Bilingual modeling technique is able to outperform in both training and
validation.

###  3.2 Results

The evaluation of all BERT based models as mentioned before is performed using
Masked Language Modeling (MLM) and Next Sentence Prediction (NSP) while the
evaluation of RoBERTa models is done using Loss and Perplexity for the MLM
task. Table 2 shows the accuracies of MLM and NSP tasks for the Monolingual,
Bilingual and Multilingual BERT based models that we have trained.

The monolingual BERT is pretrained from scratch purely on Roman Urdu while the
Multilingual BERT is trained by performing additional pretraining steps on a
pretrained Multilingual BERT. The Bilingual BERT which achieves the best
performance is trained by performing the same additional pretraining steps as
before but on a monolingual English BERT Base model by following the
Vocabulary Augmentation technique as described in section 2.1. All pretraining
cycles use the same novel corpora proposed in Section 2.3. Each of the
pretraining and additional pretraining cycles are performed for 10K steps with
a learning rate of $2e-5$ except for Monolingual training which was performed
with a learning rate of $1e-4$, the higher learning rate was necessary when
training from scratch.

The Monolingual RoBERTa is trained from scratch purely on the proposed Roman
Urdu datasets. The Bilingual RoBERTa is trained by starting from a pretrained
English model and vocabulary augmentation is again used to fuse the new
language to create a bilingual model. The evaluation results of RoBERTa based
models are shown in Table 3. It can be seen that the Bilingual modeling
approach is also able to outperform here.

From the above experiments we see that Bilingual models of Roman Urdu
significantly outperform their Monolingual and Multilingual counter parts in
similar settings. This performance is significant in the MLM task as compared
to the NSP task which also reiterates the idea in Liu et al. (2019) that NSP
is an easier task compared to MLM. The experiments conclude that the proposed
vocabulary augmentation technique can be effectively used to significantly
improve performance for Low Resource languages given that they share
vocabulary with a high resource language.

##  4 Related Work

Neural Language modeling was first proposed by Bengio et al. (2003); Collobert
and Weston (2008) who showed that the model implicitly learnt useful
representations. The technique popularly came to be known as word embeddings.
These embeddings were a leap forward in the field of NLP notably after the
introduction of techniques like word2vec Mikolov et al. (2013), GloVe
Pennington et al. (2014), fastText Joulin et al. (2017). These early
techniques were mostly context-free and a major shortcoming was that they
couldn’t handle Polysemy. This started the search for contextual embeddings.
ELMo Peters et al. (2018) and ULMFiT Howard and Ruder (2018) were the first to
achieve substantial improvements using LSTM based language models. Following
the line of work of contextual models, a different approach, GPT Radford et
al. (2018) was proposed to tackle tasks in the GLUE benchmark Wang et al.
(2018). The new approach to contextual language modelling was to replace LSTMs
entirely with then recently released Transformers Vaswani et al. (2017). GPT
used a 12-layer decoder-only transformer which was trained for 100 epochs on
the BookCorpus Zhu et al. (2015) and achieved considerable improvements as
compared to ELMo on the same NLP tasks. The hugely popular BERT was based on a
similar strategy as GPT containing 12 levels but only used the encoder part of
the Transformer and looked at sentences bidirectionally.

Many studies have been performed in the area of bilingual and multilingual
language modeling. Several works focus on mapping word representations in
multiple languages to a unified embedding space so words in different
languages can be compared. One line of work focuses on using bilingually
aligned corpora at sentence level Zou et al. (2013); Hermann and Blunsom
(2014); Luong et al. (2015) and another tries to achieves the same on document
level Vulić and Moens (2016); Levy et al. (2016). Another line of work focuses
on utilising the isomorphic structure of languages, dictionary mappings and
shared vocabulary for ad-hoc mappings between languages Artetxe et al. (2018);
Faruqui and Dyer (2014).

##  5 Conclusion

In this research we propose a large Roman Urdu corpus with 3M sentences in
addition to a novel collection of corpora cleaned and transliterated from
Urdu. The corpora cover a variety of domains from news, IMDB reviews to
tweets. In addition to this we introduce a novel vocabulary augmentation
method that enables adding new languages to pretrained models. We also show
how this new method can be used for cross-lingual transfer and improving
performance for Low Resource languages. We have also shown that the proposed
Bilingual language modeling is able to outperform multilingual modeling given
a code-switched language like Roman Urdu. In the future we would like to
better evaluate these Roman Urdu Language models on downstream NLP tasks which
can be done by producing a GLUE like benchmark for Roman Urdu. We hope that
the corpora and methods proposed in this paper will enable the NLP community
to produce better language models especially for Low resource and resource
starved Languages like Roman Urdu.

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# Constraints on intersection forms of spin 4-manifolds bounded by Seifert
rational homology 3-spheres in terms of $\overline{\mu}$ and $\kappa$
invariants.

Masaaki Ue  Kyoto University, Kyoto, 606-8502, Japan
<EMAIL_ADDRESS>
###### Abstract.

We give some constraints on intersection forms of spin 4-manifolds bounded by
Seifert rational homology 3-spheres in terms of $\overline{\mu}$-invariant,
and compare them with those in terms of $\kappa$-invariant given by Manolescu.
Furthermore in case of a Seifert rational homology 3-sphere, we show that the
difference between $\kappa$ and $-\overline{\mu}$ is at most 2 (and there is
an example where the difference is 2), and they coincide with each other under
some extra conditions, including Floer $K_{G}$ split cases.

Let $Y$ be a closed 3-manifold with spin structure $\mathfrak{s}$. Then
$(Y,\mathfrak{s})$ bounds a compact smooth spin 4-manifold
$(W,\mathfrak{s}_{W})$, where $s_{W}$ is a spin structure of $W$ which
coincides with $\mathfrak{s}$ on $Y$. In the case where $Y$ is a Seifert (or
more generally a plumbed) rational homology 3-sphere, there is a lift
$\overline{\mu}(Y,\mathfrak{s})$ of the Rokhlin invariant
$\mu(Y,\mathfrak{s})$, which is called the $\overline{\mu}$-invariant defined
by Neumann ([15]) and Siebenmann ([22]).

In this paper we give the following constraints on the intersection forms of
compact spin 4-manifolds bounded by Seifert rational homology 3-spheres in
terms of the $\overline{\mu}$ invariant. In this paper we only consider
Seifert fibrations over orientable base 2-orbifolds.

######  Theorem 1.

Let $(W,\mathfrak{s}_{W})$ be a compact spin 4-manifold bounded by a Seifert
rational homology 3-sphere with spin structure $(Y,\mathfrak{s})$. Then
$\overline{\mu}(Y,\mathfrak{s})\equiv\sigma(W)/8\pmod{2\mathbf{Z}}$ and

$-b_{2}^{-}(W)+\frac{\sigma(W)}{8}\leq\overline{\mu}(Y,\mathfrak{s})\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8}$

where $b_{2}^{+}(W)$ (resp. $b_{2}^{-}(W)$) is the dimension of the maximal
positive definite (resp. negative definite) subspace of the intersection form
of $W$, and $\sigma(W)$ is the signature of $W$.

There are similar constraints on the intersection form of $W$ in terms of the
$\kappa$-invariant $\kappa(Y,\mathfrak{s})$ of $(Y,\mathfrak{s})$ proved by
Manolescu (which are valid for general rational homology 3-spheres). In a
certain case $\kappa(Y,\mathfrak{s})$ coincides with
$-\overline{\mu}(Y,\mathfrak{s})$, but they are not necessarily the same even
in the case of Brieskorn homology 3-spheres ([14]). However by comparing the
constraints in terms of $\kappa(Y,\mathfrak{s})$ with the estimates given by
$\overline{\mu}(Y,\mathfrak{s})$, we obtain the following result. If $Y$ is an
integral homology 3-sphere, we denote these invariants simply by $\kappa(Y)$
and $\overline{\mu}(Y)$ since $Y$ has a unique spin structure.

######  Theorem 2.

Let $(Y,\mathfrak{s})$ be a Seifert rational homology 3-sphere over a base
2-orbifold of genus 0 with spin structure.

  1. (1)

We have the following inequality.

$0\leq\kappa(Y,\mathfrak{s})+\overline{\mu}(Y,\mathfrak{s})\leq 2.$

  2. (2)

If $(Y,\mathfrak{s})$ is Floer $K_{G}$ split, then
$\kappa(Y,\mathfrak{s})+\overline{\mu}(Y,\mathfrak{s})=0$.

  3. (3)

We have $0\leq\kappa(Y,\mathfrak{s})+\kappa(-Y,\mathfrak{s})\leq 4$ (although
$\kappa(Y,\mathfrak{s})+\kappa(-Y,\mathfrak{s})$ is not necessarily $0$).

  4. (4)

If one of the multiplicities of the singular fibers for the Seifert fibration
of $Y$ is even and $\deg Y>0$, then
$\kappa(Y,\mathfrak{s})=-\overline{\mu}(Y,\mathfrak{s})$.

  5. (5)

If $Y$ is a Seifert integral homology 3-sphere (with the unique spin
structure), $\kappa(Y)+\overline{\mu}(Y)=0$ or $2$. The same claim holds for
$-Y$, and hence $\kappa(Y)+\kappa(-Y)=0$, $2$, or $4$.

Theorem 2 (4) can be extended to the case when all multiplicities are odd
under certain extra conditions on $(Y,\mathfrak{s})$ (Remark 5).

##  1\. The Fukumoto-Furuta invariant and the proof of Theorem 1.

The proof of Theorem 1 is based on the estimates of the Fukumoto-Furuta
invariant ([4]) and its relation to the $\overline{\mu}$-invariant of a
Seifert rational homology 3-sphere, which we recall in this section.

######  Definition 1.

Let $(X,\mathfrak{s}_{X})$ be a compact spin 4-orbifold with spin structure
$\mathfrak{s}_{X}$ bounded by a rational homology 3-sphere $(Y,\mathfrak{s})$
with spin structure. Choose a compact spin 4-manifold
$(X^{\prime},\mathfrak{s}_{X^{\prime}})$ with
$\partial(X^{\prime},\mathfrak{s}_{X^{\prime}})=(Y,\mathfrak{s})$ and put
$(Z,\mathfrak{s}_{Z})=(X\cup(-X^{\prime}),\mathfrak{s}_{X}\cup\mathfrak{s}_{X^{\prime}})$.
Then the Fukumoto-Furuta invariant $w(Y,X,\mathfrak{s}_{X})$ is defined to be

$w(Y,X,\mathfrak{s}_{X})=-\operatorname{ind}\mathcal{D}_{Z}(\mathfrak{s}_{Z})+\frac{\sigma(X^{\prime})}{8}$

where $\mathcal{D}_{Z}(\mathfrak{s}_{Z})$ is the Dirac operator of
$(Z,\mathfrak{s}_{Z})$. The value of $w(Y,X,\mathfrak{s}_{X})$ does not depend
on the choice of $X^{\prime}$.

We note that our sign convention of $w(Y,X,\mathfrak{s}_{X})$ is opposite to
the original definition by Fukumoto-Furuta [4] and [5]. Let $\mathrm{Sing}X$
be the set of all singularities of $X$. Hereafter we only consider 4-orbifolds
with isolated singularities. In such cases the link of every
$x\in\mathrm{Sing}X$ is a spherical 3-manifold with spin structure induced
from $\mathfrak{s}_{X}$. Then the regular neighborhood of
$x_{i}\in\mathrm{Sing}X$ in $X$ is a cone $cS_{i}$ over a spherical 3-manifold
$S_{i}$, and
$X_{0}=X\setminus\cup_{x_{i}\in{\mathrm{S}ing}X}\operatorname{Int}cS_{i}$ is a
4-manifold with spin structure $\mathfrak{s}_{X_{0}}$ induced from
$\mathfrak{s}_{X}$ satisfying $\partial X_{0}=Y\cup\cup_{i}(-S_{i})$.

######  Proposition 1.

[5] The orbifold index theorem shows that

$\displaystyle\operatorname{ind}\mathcal{D}_{Z}(\mathfrak{s}_{Z})$
$\displaystyle=-\frac{1}{24}\int_{Z}p_{1}(Z)+\sum_{i}\delta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}}),$
$\displaystyle\sigma(Z)$
$\displaystyle=\frac{1}{3}\int_{Z}p_{1}(Z)+\sum_{i}\delta^{\mathrm{sign}}(S_{i})$

where $p_{1}(Z)$ is the first Pontrjagin class of $Z$, $\mathfrak{s}_{S_{i}}$
is the spin structure of $S_{i}$ induced from $\mathfrak{s}_{X}$,
$\delta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}})$ and
$\delta^{\mathrm{sign}}(S_{i})$ are the contributions of
$x_{i}\in\mathrm{Sing}X$ to the corresponding indices. Using these formulae
$w(Y,X,\mathfrak{s}_{X})$ is represented as follows.

$w(Y,X,\mathfrak{s}_{X})=\frac{1}{8}(\sigma(Z)+\sum_{x_{i}\in\mathrm{Sing}X}\delta_{x_{i}})+\frac{\sigma(X^{\prime})}{8}=\frac{1}{8}(\sigma(X)+\sum_{x_{i}\in\mathrm{Sing}X}\delta_{x_{i}}).$

Here we put
$\delta_{x_{i}}=-(8\delta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}})+\delta^{\mathrm{sign}}(S_{i}))$.

For later use, we reformulate the contributions of $\mathrm{Sing}X$ to the
above indices in terms of the $\eta$ invariants ([1]).

######  Proposition 2.

[26] Let $g_{i}$ be a standard metric of $S_{i}$ (which is induced from the
metric of the constant curvature on the 3-sphere of radius 1 as the universal
covering of $S_{i}$). Then we have

$\eta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}},g_{i})=2\delta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}}),\quad\eta^{\mathrm{sign}}(S_{i},g_{i})=\delta^{\mathrm{sign}}(S_{i}).$

Here $\eta^{\mathrm{Dir}}(S_{i},\mathfrak{s}_{S_{i}},g_{i})$ and
$\eta^{\mathrm{sign}}(S_{i},g_{i})$ are the $\eta$ invariants of the Dirac
operator of $(S_{i},\mathfrak{s}_{S_{i}},g_{i})$ and the sigunature operator
of $(S_{i},g_{i})$ respectively. We note that the kernel of the Dirac operator
of $(S_{i},\mathfrak{s}_{S_{i}},g_{i})$ is $0$ since $g_{i}$ is a metric of
positive scalar curvature.

The value of $w(Y,X,\mathfrak{s}_{X})$ depends on both $Y$ and $X$ in general,
but if $Y$ is a Seifert rational homology 3-sphere, it is related to the
$\overline{\mu}$ invariant of $Y$ by some particular choice of $X$.

######  Definition 2.

Let $P(\Gamma)$ be a plumbed 4-manifold associated with a weighted graph
$\Gamma$. We assume that a plumbed 3-manifold $Y=\partial P(\Gamma)$ is a
rational homology 3-sphere. Thus $\Gamma$ is a tree such that each vertex $v$
with weight $e(v)$ of $\Gamma$ corresponds to a $D^{2}$-bundle over $S^{2}$
with Euler class $e(v)$. Let $\\{e_{v}\\}$ be a basis of
$H_{2}(P(\Gamma),\mathbf{Z})$ each of which is represented by a zero section
of the $D^{2}$-bundle over $S^{2}$ corresponding to $v$. If we fix a spin
structure $\mathfrak{s}_{Y}$ of $Y=\partial P(\Gamma)$, there exists a unique
characteristic element $w_{\Gamma}=\sum_{v}\epsilon_{v}e_{v}$ satisfying the
following conditions.

  * •

$\epsilon_{v}$ is either $0$ or $1$.

  * •

The Poincaré dual $PDw_{\Gamma}\pmod{2}\in H^{2}(P(\Gamma),\partial
P(\Gamma),\mathbf{Z}_{2})$ of $w_{\Gamma}$ is a lift of $w_{2}(P(\Gamma))$,
which is an obstruction to extending $\mathfrak{s}_{Y}$ to a spin structure of
$P(\Gamma)$.

We denote by $\operatorname{supp}(\Gamma)$ the set $\\{v\in\Gamma\ |\
\epsilon_{v}=1\\}$. Then the $\overline{\mu}$ invariant of
$(Y,\mathfrak{s}_{Y})$ is defined to be

$\overline{\mu}(Y,\mathfrak{s}_{Y})=\frac{1}{8}(\sigma(P(\Gamma))-w_{\Gamma}\cdot
w_{\Gamma}).$

The value of $\overline{\mu}(Y,\mathfrak{s}_{Y})$ does not depend on $\Gamma$,
although $\Gamma$ satisfying $Y=\partial P(\Gamma)$ is not unique.

######  Remark 1.

The $\overline{\mu}$ invariant has the following properties.

  1. (1)

$\overline{\mu}(Y,\mathfrak{s}_{Y})\equiv\mu(Y,\mathfrak{s}_{Y})\mod{2\mathbf{Z}}$.
If $Y$ is a plumbed integral homology 3-sphere with the unique spin structure,
$\overline{\mu}(Y)$ is an integral lift of $\mu(Y)\in\mathbf{Z}_{2}$.

  2. (2)

If $Y$ is a Seifert rational homology 3-sphere (with base 2-orbifold of genus
$0$), then $Y$ is a plumbed 3-manifold corresponding to a star-shaped graph.

  3. (3)

In some cases the $\overline{\mu}$ invariant is related to other spin
cobordism invariants. For example, the relations between
$\overline{\mu}(Y,\mathfrak{s})$ and the correction term $d(Y,\mathfrak{s})$
of the Heegaard Floer homology ([18]) are discussed in [26], [21]. If $Y$ is a
Seifert integral homology 3-sphere, $-\overline{\mu}(Y)$ coincides with the
$\beta$-invariant $\beta(Y)$ defined by Manolescu [13] (Stoffregen [23]), and
the same equality holds if $Y$ is a plumbed rational homology 3-sphere
corresponding to a graph with at most one bad vertex (Dai [2]). In case of a
Seifert integral homology 3-sphere, the relations between $\overline{\mu}(Y)$
and the invariants $\alpha(Y)$, $\gamma(Y)$ and $d(Y)$ are also determined in
[23]. More generally,
$\underline{d}(Y,\mathfrak{s})=-2\overline{\mu}(Y,\mathfrak{s})$ for every
almost rational plumbed rational homology 3-sphere $Y$ (Dai-Manolescu [3]),
where $\underline{d}(Y,\mathfrak{s})$ is the correction term of the involutive
Heegaard Floer homology ([8]). In all above cases, the $\overline{\mu}$
invariant is a spin rational homology cobordism invariant.

The relation between the $\overline{\mu}$ invariant and the Fukumoto-Furuta
invariant for a Seifert fibration is discussed in [4], [5], [19], [24], [25].

If a Seifert rational homology 3-sphere $Y$ has a fibration over a 2-orbifold
$S^{2}(a_{1},\cdots,a_{n})$ of genus $0$ with singular points of multiplicity
$a_{i}$ $(1\leq i\leq n)$, then $Y$ is represented by Seifert invariants of
the form $\\{b,(a_{1},b_{1}),\dots,(a_{n},b_{n})\\}$ satisfying
$0<b_{i}<a_{i}$, $\gcd(a_{i},b_{i})=1$ $(1\leq i\leq n)$, and the degree of
$Y$ is defined to be $\deg Y=b+\sum_{i=1}^{n}b_{i}/a_{i}$ (we follow the
definition of [10], whose sign convention is opposite to that in [25], where
we follow the definition of [16]). In case of a Seifert rational homology
3-sphere, we have the following result.

######  Proposition 3.

[25]111The definitions of $w$ and $\overline{\mu}$ invariants in [25] and [26]
are both 8 times those in this paper. Let $(Y,\mathfrak{s})$ be a Seifert
rational homology 3-sphere with spin structure. Then there exist compact spin
4-orbifolds $(X^{\pm},\mathfrak{s}_{X^{\pm}})$ with
$\partial(X^{\pm},\mathfrak{s}_{X^{\pm}})=(Y,\mathfrak{s})$ satisfying the
following conditions.

  1. (1)

$b_{1}(X^{\pm})=0$, $b_{2}^{+}(X^{+})\leq 1$, and $b_{2}^{-}(X^{-})\leq 1$.

  2. (2)

$w(Y,X^{+},\mathfrak{s}_{X^{+}})=w(Y,X^{-},\mathfrak{s}_{X^{-}})=\overline{\mu}(Y,\mathfrak{s})$.

If $Y$ is spherical, then we can choose $X^{\pm}$ so that both of them are the
cone over $Y$. Suppose that one of the multiplicities of the singular fibers
for the Seifert fibration of $Y$ is even. Then we can choose $X^{\pm}$ so that
$X^{+}=X^{-}$, satisfying $b_{2}^{+}(X^{\pm})=1$ and $b_{2}^{-}(X^{\pm})=0$ if
$\deg Y>0$, and $b_{2}^{+}(X^{\pm})=0$ and $b_{2}^{-}(X^{\pm})=1$ if $\deg
Y<0$.

To construct $X^{\pm}$, we choose appropriate weighted graphs $\Gamma^{\pm}$
satisfying $\partial P(\Gamma^{\pm})=Y$ and subgraphs $\Gamma^{\pm}_{0}$ of
$\Gamma^{\pm}$ such that $\Gamma^{\pm}_{0}$ contain
$\operatorname{supp}(\Gamma^{\pm})$ and each of $S^{\pm}:=\partial
P(\Gamma_{0}^{\pm})$ is a union of spherical 3 -manifolds (in fact lens
spaces). Then embed $P(\Gamma^{\pm}_{0})$ in
$\operatorname{Int}P(\Gamma^{\pm})$ and replace their images with the cones
over $S^{\pm}$ to obtain $X^{\pm}$. Such $X^{\pm}$ are constructed by Saveliev
[19] for Seifert integral homology 3-spheres, and the result is extended to
the case of Seifert rational homology 3-spheres in [25].

The proof of Theorem 1 is based on Proposition 3 and the following orbifold
$10/8$-theorem.

######  Theorem 3.

[4] Let $(Z,\mathfrak{s}_{Z})$ be a closed spin 4-orbifold with spin structure
$\mathfrak{s}_{Z}$. Then

  1. (1)

$\operatorname{ind}\mathcal{D}_{Z}(\mathfrak{s}_{Z})\equiv 0\pmod{2}$,

  2. (2)

$\operatorname{ind}\mathcal{D}_{Z}(\mathfrak{s}_{Z})=0$ or otherwise

$1-b_{2}^{-}(Z)\leq\operatorname{ind}\mathcal{D}_{Z}(\mathfrak{s}_{Z})\leq
b_{2}^{+}(Z)-1.$

For a compact spin manifold $(W,\mathfrak{s}_{W})$ with
$\partial(W,\mathfrak{s}_{W})=(Y,\mathfrak{s})$, let
$(Z^{\pm},\mathfrak{s}_{Z^{\pm}})=(X^{\pm},\mathfrak{s}_{X^{\pm}})\cup(-W,\mathfrak{s}_{W})$
be closed spin 4-orbifolds where $X^{\pm}$ are the 4-orbifolds chosen in
Proposition 3. Then applying Theorem 3 to $(Z^{\pm},\mathfrak{s}_{Z^{\pm}})$
we have
$\overline{\mu}(Y,\mathfrak{s})\equiv\frac{\sigma(W)}{8}\pmod{2\mathbf{Z}}$
and

$\displaystyle\overline{\mu}(Y,\mathfrak{s})$
$\displaystyle=w(Y,X^{-},\mathfrak{s}_{X^{-}})=-\operatorname{ind}\mathcal{D}_{Z^{-}}(\mathfrak{s}_{Z^{-}})+\frac{\sigma(W)}{8}\leq
b_{2}^{-}(Z^{-})-1+\frac{\sigma(W)}{8}$
$\displaystyle=b_{2}^{-}(X^{-})+b_{2}^{+}(W)-1+\frac{\sigma(W)}{8}\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8},$
$\displaystyle\overline{\mu}(Y,\mathfrak{s})$
$\displaystyle=w(Y,X^{+},\mathfrak{s}_{X^{+}})=-\operatorname{ind}\mathcal{D}_{Z^{+}}(\mathfrak{s}_{Z^{+}})+\frac{\sigma(W)}{8}\geq
1-b_{2}^{+}(Z^{+})+\frac{\sigma(W)}{8}$
$\displaystyle=1-b_{2}^{-}(W)-b_{2}^{+}(X^{+})+\frac{\sigma(W)}{8}\geq-
b_{2}^{-}(W)+\frac{\sigma(W)}{8}.$

The above inequalities hold even if
$\operatorname{ind}\mathcal{D}_{Z^{\pm}}(\mathfrak{s}_{Z^{\pm}})=0$. The
second inequality is also deduced from the first one with respect to $X^{\pm}$
chosen for $-Y$ and $-W$, since $\overline{\mu}(-Y)=-\overline{\mu}(Y)$. This
completes the proof of Theorem 1. We will use Proposition 3 to prove Theorem 2
in §3.

######  Remark 2.

In some cases we obtain slightly better estimates than those in Theorem 1 by
the above inequalities. If $Y$ is a spherical 3-manifold $S$, we can choose
$X^{\pm}$ so that $X^{+}=X^{-}=cS$ and hence applying Theorem 3 to
$Z=cS\cup(-W)$ we have either

$1-b_{2}^{-}(W)+\frac{\sigma(W)}{8}\leq\overline{\mu}(S,\mathfrak{s})\leq
b_{2}^{+}(W)-1+\frac{\sigma(W)}{8}$

and
$\overline{\mu}(S,\mathfrak{s})\equiv\frac{\sigma(W)}{8}\pmod{2\mathbf{Z}}$,
or $\overline{\mu}(S,\mathfrak{s})=\frac{\sigma(W)}{8}$.

If $Y$ has a fibration over $S^{2}(a_{1},\dots,a_{n})$ and one of $a_{i}$ is
even, then we can choose $X^{\pm}$ so that $X^{+}=X^{-}$ as in Proposition 3.
If $\deg Y>0$, $X^{\pm}$ satisfies $b_{2}^{+}(X^{\pm})=1$,
$b_{2}^{-}(X^{\pm})=0$, and hence we have either

$-b_{2}^{-}(W)+\frac{\sigma(W)}{8}\leq\overline{\mu}(Y,\mathfrak{s})\leq
b_{2}^{+}(W)-1+\frac{\sigma(W)}{8}$

or $\overline{\mu}(Y,\mathfrak{s})=\frac{\sigma(W)}{8}$. If $\deg Y<0$, then
since $X^{\pm}$ satisfies $b_{2}^{+}(X^{\pm})=0$, $b_{2}^{-}(X^{\pm})=1$, we
have either

$1-b_{2}^{-}(W)+\frac{\sigma(W)}{8}\leq\overline{\mu}(Y,\mathfrak{s})\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8}$

or $\overline{\mu}(Y,\mathfrak{s})=\frac{\sigma(W)}{8}$. In either case we
have
$\overline{\mu}(Y,\mathfrak{s})\equiv\frac{\sigma(W)}{8}\pmod{2\mathbf{Z}}$.
If $Y$ is a Seifert integral homology 3-sphere, then $Y$ is uniquely
determined by $a_{i}$’s up to orientation, and is denoted by
$\Sigma(a_{1},\dots,a_{n})$ if $\deg Y<0$. Then $\Sigma(a_{1},\dots,a_{n})$
bounds an associated orbifold disk bundle $D(a_{1},\dots,a_{n})$ over
$S^{2}(a_{1},\dots,a_{n})$. If one of $a_{i}$ is even, then
$D(a_{1},\dots,a_{n})$ is spin and $b_{2}^{-}(D(a_{1},\dots,a_{n}))=1$,
$b_{2}^{+}(D(a_{1},\dots,a_{n})=0$ ([4]). Hence putting
$X^{\pm}=D^{2}(a_{1},\dots,a_{n})$ we also obtain the same inequality. But in
other cases $D(a_{1},\dots,a_{n})$ is not necessarily spin, so we cannot use
it.

##  2\. The $\kappa$ invariant for a rational homology 3-sphere.

For a rational homology 3-sphere, $K$-theoretic invariants are defined via the
theory of the $\operatorname{Pin}(2)$ Seiberg-Witten-Floer stable homotopy
type ([11], [12]) in [6], [14]. In this section we recall the definition of
Manolescu’s $\kappa$ invariant for a rational homology 3-sphere with spin
structure $(Y,\mathfrak{s})$ and its properties [14] 222Theorems 4, 5, and
Corollary 1 below are described for integral homology 3-spheres in [14]. But
as is pointed out in [14], these results are easily generalized to the case of
rational homology 3-spheres. We give the outline of their proofs below.
Hereafter we put $G=\operatorname{Pin}(2)=S^{1}\cup jS^{1}$.

######  Definition 3.

[14]

  1. (1)

Let $\widetilde{\mathbf{R}}$ (resp. $\widetilde{\mathbf{C}}$) be a real (resp.
complex) 1-dimensional $G$\- representation space where $G$ acts as
multiplication by $\pm 1$ via the natural projection $G\to G/S^{1}\cong\\{\pm
1\\}$, and let $\mathbf{H}$ be the set of quaternions on which $G$ acts as
left multiplication. Then the representation ring $R(G)$ is generated by
$z=2-\mathbf{H}$ and $w=1-\widetilde{\mathbf{C}}$ with relations
$w^{2}=wz=2w$.

  2. (2)

A compact $G$-space $X$ is called a space of type SWF at level $s$ if the
fixed point set $X^{S^{1}}$ of the action of $S^{1}\subset G$ on $X$ is $G$
homotopy equivalent to $(\widetilde{\mathbf{R}}^{s})^{+}$ and $G$ acts freely
on $X\setminus X^{S^{1}}$.

######  Definition 4.

[14] Let $X$ be a space of type SWF at level $2t$ (in which case $X^{S^{1}}$
is $G$ homotopy equivalent to $(\widetilde{\mathbf{C}}^{t})^{+}$). Then the
inclusion $\iota:X^{S^{1}}\to X$ induces the map
$\iota^{*}:\widetilde{K}_{G}(X)\to\widetilde{K}_{G}(X^{S^{1}})$ whose image is
represented as $\mathcal{J}(X)\cdot\beta_{(\widetilde{\mathbf{C}}^{t})^{+}}$,
where $\mathcal{J}(X)$ is an ideal of $R(G)\cong\widetilde{K}_{G}(S^{0})$ and
$\beta_{(\widetilde{\mathbf{C}}^{t})^{+}}$ is the Bott class of
$\widetilde{K}_{G}((\widetilde{\mathbf{C}}^{t})^{+})$.

  1. (1)

There exists $k\geq 0$ such that both $w^{k}$and $z^{k}$ belong to
$\mathcal{J}(X)$, so we can define

$k(X)=\min\\{k\geq 0\ |\ \text{there exists $x\in\mathcal{J}(X)$ such that
$wx=z^{k}w$}\\}$

  2. (2)

$X$ is called $K_{G}$ split if $\mathcal{J}(X)=(z^{k})$ for some $k\geq 0$.

######  Lemma 1.

[14]

  1. (1)

Let $f:X\to X^{\prime}$ is a $G$ equivariant map between two spaces of type
SWF at level $2t$, and assume that
$f^{S^{1}}:X^{S^{1}}\to{X^{\prime}}^{S^{1}}$ is a $G$ homotopy equivalence,
where $f^{S^{1}}$ is the restriction of $f$ to $X^{S^{1}}$. Then $k(X)\leq
k(X^{\prime})$.

  2. (2)

Let $X$ and $X^{\prime}$ be spaces of type SWF at level $2t$ and $2t^{\prime}$
respectively and $t<t^{\prime}$. Let $f:X\to X^{\prime}$ be a $G$ equivariant
map such that $f^{G}:X^{G}\to{X^{\prime}}^{G}$ is a homotopy equivalence,
where $f^{G}$ is the restriction of $f$ to the fixed point set of the $G$
action on $X$. Then $k(X)+t\leq k(X^{\prime})+t^{\prime}$. Furthermore, if $X$
is $K_{G}$ split, then $k(X)+t+1\leq k(X^{\prime})+t^{\prime}$.

A space of type SWF associated with a rational homology 3-sphere with spin
structure $(Y,\mathfrak{s})$ is constructed as follows. We fix a Riemann
metric $g$ of $Y$. Let $V=i\ker d^{*}\oplus\Gamma(S)$ where $i\ker
d^{*}\subset i\Omega^{1}(Y)$ and $\Gamma(S)$ is the space of sections of the
Spinor bundle $S$ over $Y$ associated with $\mathfrak{s}$. Then the action of
$G$ on $V$ is defined as

$e^{i\theta}(a,\phi)=(a,e^{i\theta}\phi),\quad j(a,\phi)=(-a,j\phi),$

and the gradient of the Chern-Simons-Dirac functional $\nabla CSD$ over $V$ is
represented as a sum of a $G$ equivariant linear operator $\ell$ and a compact
operator $c$. Let $V^{\mu}_{\tau}$ be the finite dimensional subspace of $V$
spanned by eigenvectors corresponding to the eigenvalues of $\ell$ belonging
to $(\tau,\mu]$. Then $V^{\mu}_{\tau}$ is a direct sum of
$V^{\mu}_{\tau}(\widetilde{\mathbf{R}})=\widetilde{\mathbf{R}}^{s}$ and
$V^{\mu}_{\tau}(\mathbf{H})=\mathbf{H}^{t}$ for some $s$, $t$. Manolescu
defined a finite dimensional approximation of $\nabla CSD$ over
$V^{\nu}_{-\nu}$ and corresponding flows (approximated Seiberg-Witten flows)
for a large $\nu>0$ and showed that a set $S_{\nu}$ of critical points and
flows between them in a large ball centered at the origin in $V^{\nu}_{-\nu}$
forms an isolated invariant set. Thus a $G$ equivariant Conley index $I_{\nu}$
of $S_{\nu}$ is defined up to $G$ homotopy equivalence ([11], [13], [14]).

######  Proposition 4.

[13], [14]

  1. (1)

There exists a unique critical point $\Theta$ of the above flow corresponding
to the unique isolated and nondegenerate reducible critical point of $\nabla
CSD$ (which is perturbed if necessary). The Conley index $I(\Theta)$ of
$\Theta$ is $(V^{0}_{-\nu})^{+}$.

  2. (2)

$I_{\nu}$ is a space of type SWF at level $\dim
V^{0}_{-\nu}(\widetilde{\mathbf{R}})$.

######  Definition 5.

[13] For $(Y,\mathfrak{s})$ choose a compact 4-manifold with spin structure
$(W,\mathfrak{s}_{W})$ bounded by $(Y,\mathfrak{s})$ and a metric $g_{W}$ of
$W$ extending the given metric $g$ of $Y$. Then define $n(Y,\mathfrak{s},g)$
to be

$n(Y,\mathfrak{s},g)=\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})+\frac{\sigma(W)}{8}\in\frac{1}{8}\mathbf{Z}$

where $\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})$ is
the Atiyah-Patodi-Singer index of the Dirac operator of $(W,\mathfrak{s}_{W})$
associated with $g_{W}$. The value of $n(Y,\mathfrak{s},g)$ is independent of
the choice of $(W,\mathfrak{s}_{W})$. If $Y$ is an integral homology 3-sphere,
$n(Y,\mathfrak{s},g)\in\mathbf{Z}$ since $\sigma(W)$ is divisible by 8.

######  Definition 6.

[11] [13] [14] A Seiberg-Witten Floer stable homotopy type
$SWF(Y,\mathfrak{s})$ of $(Y,\mathfrak{s})$ is a formal desuspension of
$I_{\nu}$ (up to stable equivalence) defined as follows: If $\dim
V^{0}_{-\nu}(\widetilde{\mathbf{R}})$ is even, then
$V^{0}_{-\nu}=\widetilde{\mathbf{R}}^{2k}\oplus\mathbf{H}^{\ell}$ for some
$k$, $\ell$ and

$SWF(Y,\mathfrak{s})=\Sigma^{-(k\widetilde{\mathbf{C}}+(\ell+\frac{1}{2}n(Y,\mathfrak{s},g))\mathbf{H})}I_{\nu},$

while if $\dim V^{0}_{\\-nu}(\widetilde{\mathbf{R}})$ is odd, then
$V^{0}_{-\nu}=\widetilde{\mathbf{R}}^{2k-1}\oplus\mathbf{H}^{\ell}$ for some
$k$, $\ell$ and

$SWF(Y,\mathfrak{s})=\Sigma^{-(k\widetilde{\mathbf{C}}+(\ell+\frac{1}{2}n(Y,\mathfrak{s},g))\mathbf{H})}(\Sigma^{\widetilde{\mathbf{R}}}I_{\nu}).$

More precisely, $SWF(Y,\mathfrak{s})$ is an object of the $G$ equivariant
graded suspension category $\mathfrak{C}$.

######  Definition 7.

[14] The $\kappa$ invariant of $(Y,\mathfrak{s})$ is defined to be

$\displaystyle\kappa(Y,\mathfrak{s})$ $\displaystyle=2k(SWF(Y,\mathfrak{s}))$
$\displaystyle=\begin{cases}2k(I_{\nu})-2\dim_{\mathbf{H}}V^{0}_{-\nu}(\mathbf{H})-n(Y,\mathfrak{s},g)&\text{if
$I_{\nu}$ is at even level, }\\\
2k(\Sigma^{\widetilde{\mathbf{R}}}I_{\nu})-2\dim_{\mathbf{H}}V^{0}_{-\nu}(\mathbf{H})-n(Y,\mathfrak{s},g)&\text{if
$I_{\nu}$ is at odd level}.\end{cases}$

######  Proposition 5.

[14]

  1. (1)

$\kappa(Y,\mathfrak{s})\equiv\overline{\mu}(Y,\mathfrak{s})\pmod{2\mathbf{Z}}$.
If $(Y,\mathfrak{s})$ is an integral homology 3-sphere with the unique spin
structure, then $\kappa(Y,\mathfrak{s})$ is an integral lift of
$\overline{\mu}(Y,\mathfrak{s})\in\mathbf{Z}_{2}$.

  2. (2)

$\kappa(Y,\mathfrak{s})$ is a spin rational homology cobordism invariant.

For a spin 4-manifold $(W,\mathfrak{s}_{W})$ bounded by $(Y,\mathfrak{s})$,
$\ker\mathcal{D}_{W}(\mathfrak{s}_{W})$ and
$\operatorname{coker}\mathcal{D}_{W}(\mathfrak{s}_{W})$ are vector spaces over
$\mathbf{H}$, and
$\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})=2\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})$.
It follows that $\kappa(Y,\mathfrak{s})\equiv
n(Y,\mathfrak{s},g)\equiv\sigma(W)/8\equiv\mu(Y,\mathfrak{s})\pmod{2\mathbf{Z}}$.
The second claim in Proposition 5 is proved by applying Theorem 4 below to a
spin rational homology cobordism between rational homology 3-spheres.

######  Theorem 4.

[14]

Let $(Y_{i},\mathfrak{s}_{i})$ $(i=0,1)$ be rational homology 3-spheres with
spin structures and $(W,\mathfrak{s}_{W})$ be a spin cobordism with
$b_{1}(W)=0$ from $(Y_{0},\mathfrak{s}_{0})$ to $(Y_{1},\mathfrak{s}_{1})$.
Then

$-b_{2}^{-}(W)+\frac{\sigma(W)}{8}-1\leq\kappa(Y_{0},\mathfrak{s}_{0})-\kappa(Y_{1},\mathfrak{s}_{1})\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8}+1.$

The first term of the above inequality can be replaced by
$-b_{2}^{-}(W)+\frac{\sigma(W)}{8}$ if $b_{2}^{-}(W)$ is even, and the last
term can be replaced by $b_{2}^{+}(W)+\frac{\sigma(W)}{8}$ if $b_{2}^{+}(W)$
is even.

######  Definition 8.

[14] $(Y,\mathfrak{s})$ is called Floer $K_{G}$ split if $I_{\nu}$ (resp.
$\Sigma^{\widetilde{\mathbf{R}}}I_{\nu}$) is $K_{G}$ split when $I_{\nu}$ is
at even level (resp. at odd level), where $I_{\nu}$ is the Conley index that
appears in the definition of $\kappa(Y,\mathfrak{s})$.

If one of $(Y_{i},\mathfrak{s}_{i})$ is Floer $K_{G}$ split, the inequalities
in Theorem 4 are slightly improved.

######  Theorem 5.

[14] Let $(W,\mathfrak{s}_{W})$ be a spin cobordism as in Theorem 4.

  1. (1)

If $(Y_{0},\mathfrak{s}_{0})$ is Floer $K_{G}$ split, then

$\displaystyle\kappa(Y_{0},\mathfrak{s}_{0})-\kappa(Y_{1},\mathfrak{s}_{1})$
$\displaystyle\leq\begin{cases}b_{2}^{+}(W)+\frac{\sigma(W)}{8}-1&\quad\text{if
$b_{2}^{+}(W)$ is odd},\\\ b_{2}^{+}(W)+\frac{\sigma(W)}{8}-2&\quad\text{if
$b_{2}^{+}(W)$ is even and $b_{2}^{+}(W)>0$. }\end{cases}$

  2. (2)

If $(Y_{1},\mathfrak{s}_{1})$ is Floer $K_{G}$ split, then

$\displaystyle\kappa(Y_{1},\mathfrak{s}_{1})-\kappa(Y_{0},\mathfrak{s}_{0})$
$\displaystyle\leq\begin{cases}b_{2}^{-}(W)-\frac{\sigma(W)}{8}-1&\quad\text{if
$b_{2}^{-}(W)$ is odd, }\\\ b_{2}^{-}(W)-\frac{\sigma(W)}{8}-2&\quad\text{if
$b_{2}^{-}(W)$ is even and $b_{2}^{-}(W)>0$. }\end{cases}$

Since $S^{3}$ with the unique spin structure is Floer $K_{G}$ split and
$\kappa(S^{3})=0$, we have the following corollary.

######  Corollary 1.

[14]

Let $(Y,\mathfrak{s})$ be a rational homology 3-sphere with spin structure and
$(W,\mathfrak{s}_{W})$ be a compact spin 4-manifold with $b_{1}(W)=0$ and
$\partial(W,\mathfrak{s}_{W})=(Y,\mathfrak{s})$.

  1. (1)

$-\kappa(Y,\mathfrak{s})\leq\begin{cases}b_{2}^{+}(W)+\frac{\sigma(W)}{8}-1&\quad\text{if
$b_{2}^{+}(W)$ is odd,}\\\ b_{2}^{+}(W)+\frac{\sigma(W)}{8}-2&\quad\text{if
$b_{2}^{+}(W)$ is even and $b_{2}^{+}(W)>0$ .}\end{cases}$

  2. (2)

Furthermore if $(Y,\mathfrak{s})$ is Floer $K_{G}$ split, then we have

$\kappa(Y,\mathfrak{s})\leq\begin{cases}b_{2}^{-}(W)-\frac{\sigma(W)}{8}-1&\quad\text{if
$b_{2}^{-}(W)$ is odd ,}\\\ b_{2}^{-}(W)-\frac{\sigma(W)}{8}-2&\quad\text{if
$b_{2}^{-}(W)$ is even and $b_{2}^{-}(W)>0$. }\end{cases}$

######  Remark 3.

Theorem 5 does not hold for a cobordism $W$ with $b_{2}^{+}(W)=0$ or
$b_{2}^{-}(W)=0$. Likewise for a spin 4-manifold $(W,\mathfrak{s}_{W})$
bounded by $(Y,\mathfrak{s})$ with $b_{2}^{+}(W)=0$ (resp. $b_{2}^{-}(W)=0$),
we only have the inequality $-\kappa(Y,\mathfrak{s})\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8}$ (resp. $\kappa(Y,\mathfrak{s})\leq
b_{2}^{-}(W)-\frac{\sigma(W)}{8}$) (as in the case where $W=S^{3}\times[0,1]$
or $W=D^{4}$).

The proofs of Theorems 4 and 5 are based on the finite dimensional
approximation of the Seiberg-Witten map for a cobordism $W$ with certain
boundary conditions developed in [11]. Let $(V_{i})^{\tau}_{\mu}$ be the
finite dimensional vector spaces used in the definitions of the approximated
Seiberg-Witten flows for $(Y_{i},\mathfrak{s}_{i})$, and $I_{i}$ be the Conley
indices $I_{\nu}$ used in the definitions of $SWF(Y_{i},\mathfrak{s}_{i})$
$(i=0,1)$. Then we obtain a $G$ equivariant map $f:Z_{0}\to Z_{1}$ where
$Z_{0}$ and $Z_{1}$ are defined (up to stable equivalence) as follows ([14]).

$\displaystyle
Z_{0}=(\mathbf{H}^{N+\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})-\dim(V_{0})^{0}_{-\nu}(\mathbf{H})})^{+}\wedge(\widetilde{\mathbf{R}}^{M-\dim_{\mathbf{R}}(V_{0})^{0}_{-\nu}(\widetilde{\mathbf{R}})})^{+}\wedge
I_{0}$ $\displaystyle
Z_{1}=(\mathbf{H}^{N-\dim_{\mathbf{H}}(V_{1})^{0}_{-\nu}(\mathbf{H})})^{+}\wedge(\widetilde{\mathbf{R}}^{M-\dim_{\mathbf{R}}(V_{1})^{0}_{-\nu}(\widetilde{\mathbf{R}})+b_{2}^{+}(W)})^{+}\wedge
I_{1}$

for some $M$, $N$ (we may assume that $M$ is even). Here $Z_{0}$ and $Z_{1}$
are spaces of type SWF at level $M$ and $M+b_{2}^{+}(W)$ respectively, and the
restriction of $f$ to the $S^{1}$ fixed point sets induces the map

$f^{S^{1}}:Z_{0}^{S^{1}}\cong S^{M}\to
Z_{1}^{S^{1}}\cong(\widetilde{\mathbf{R}}^{b_{2}^{+}(W)})^{+}\wedge S^{M}$

which is a $G$ homotopy equivalence if $b_{2}^{+}(W)=0$. Furthermore the
restriction of $f$ to the $G$ fixed point sets induces a $G$ homotopy
equivalence $f^{G}:Z_{0}^{G}\to Z_{1}^{G}$. Hence if $b_{2}^{+}(W)$ is even,
by Lemma 1 we obtain

$k(Z_{0})\leq k(Z_{1})+\frac{1}{2}b_{2}^{+}(W).$

If $I_{i}$ is at odd level, then $I_{i}$ and $M$ in the description of $Z_{i}$
are replaced with $\Sigma^{\widetilde{\mathbf{R}}}I_{i}$ and $M-1$
respectively. Then we have

$\displaystyle
k(Z_{0})=\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})-\dim_{\mathbf{H}}(V_{0})^{0}_{-\nu}(\mathbf{H})+k(I_{0}^{\prime})+N$
$\displaystyle
k(Z_{1})=-\dim_{\mathbf{H}}(V_{1})^{0}_{-\nu}(\mathbf{H})+k(I_{1}^{\prime})+N$

where $I_{i}^{\prime}=I_{i}$ (resp. $\Sigma^{\widetilde{\mathbf{R}}}I_{i}$) if
$I_{i}$ is at even level (resp. odd level), since
$k(\Sigma^{\widetilde{\mathbf{C}}}X)=k(X)$ and
$k(\Sigma^{\mathbf{H}}X)=k(X)+1$ for any space $X$ of type SWF at even level
([14]). It follows from the above inequality and the definition of
$\kappa(Y_{i},\mathfrak{s}_{i})$ that

$\displaystyle\kappa(Y_{0},\mathfrak{s}_{0})-\kappa(Y_{1},\mathfrak{s}_{1})$
$\displaystyle\leq
b_{2}^{+}(W)-\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})+n(Y_{1},\mathfrak{s}_{1},g_{1})-n(Y_{0},\mathfrak{s}_{0},g_{0})$
$\displaystyle=b_{2}^{+}(W)+\frac{\sigma(W)}{8}$

since
$n(Y_{1},\mathfrak{s}_{1},g_{1})-n(Y_{0},\mathfrak{s}_{0},g_{0})=\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})+\frac{\sigma(W)}{8}$
(Here we assume that $\ker\mathcal{D}_{Y_{i}}(\mathfrak{s}_{i})=0$ by using
the perturbed Chern-Simons-Dirac functional if necessary, which ensures that
$n(-Y_{i},\mathfrak{s}_{i},g_{i})=-n(Y_{i},\mathfrak{s}_{i},g_{i})$.) If
$b_{2}^{+}(W)$ is odd, by using $W\sharp S^{2}\times S^{2}$ instead of $W$ we
deduce

$\kappa(Y_{0},\mathfrak{s}_{0})-\kappa(Y_{1},\mathfrak{s}_{1})\leq
b_{2}^{+}(W)+\frac{\sigma(W)}{8}+1.$

We obtain the other inequalities in Theorem 4 by applying the above argument
to the cobordism $(-W,\mathfrak{s}_{W})$ from $(Y_{1},\mathfrak{s}_{1})$ to
$(Y_{0},\mathfrak{s}_{0})$. If $(Y_{0},\mathfrak{s}_{0})$ is Floer $K_{G}$
split and $b_{2}^{+}(W)$ is even and nonzero, then Lemma 1 shows that
$k(Z_{0})+1\leq k(Z_{1})+\frac{b_{2}^{+}(W)}{2}$, from which we deduce

$\displaystyle\kappa(Y_{0},\mathfrak{s}_{0})-\kappa(Y_{1},\mathfrak{s}_{1})$
$\displaystyle\leq
b_{2}^{+}(W)-2-\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})$
$\displaystyle+n(Y_{1},\mathfrak{s}_{1},g_{1})-n(Y_{0},\mathfrak{s}_{0},g_{0})=b_{2}^{+}(W)+\frac{\sigma(W)}{8}-2.$

We obtain the other inequalities in Theorem 5 by replacing $W$ with $W\sharp
S^{2}\times S^{2}$ or $-W$ as before.

##  3\. Proof of Theorem 2

To prove Theorem 2 we first consider the relation between the $\kappa$ and the
$\overline{\mu}$ invariants of a spherical 3-manifold.

######  Proposition 6.

[14] Let $(Y,\mathfrak{s})$ be a rational homology 3-sphere with spin
structure that admits a metric $g$ of positive scalar curvature. Then
$\kappa(Y,\mathfrak{s})=-n(Y,\mathfrak{s},g)$ and $(Y,\mathfrak{s})$ is Floer
$K_{G}$ split.

In the above case, the critical point set of the approximated Seiberg-Witten
flow on $V$ only consists of the unique reducible element. Hence we have
$I_{\nu}=(V^{0}_{-\nu})^{+}$ and
$\kappa(Y,\mathfrak{s})=k(S^{0})-n(Y,\mathfrak{s},g)=-n(Y,\mathfrak{s},g)$.
Furthermore the ideal $\mathfrak{J}(I_{\nu})$ (or
$\mathfrak{J}(\Sigma^{\widetilde{\mathbf{R}}}I_{\nu})$ if $I_{\nu}$ is at odd
level) is generated by $z^{k}$ for some $k$, since
$\mathfrak{J}(\Sigma^{\widetilde{\mathbf{C}}}X)=\mathfrak{J}(X)$ and
$\mathfrak{J}(\Sigma^{\mathbf{H}}X)=z\mathfrak{J}(X)$ for a space $X$ of type
SWF at even level ([14]). Hence $(Y,\mathfrak{s})$ is Floer $K_{G}$ split.

For a rational homology 3-sphere $(Y,\mathfrak{s})$ with metric $g$, let
$(W,\mathfrak{s}_{W})$ be a compact spin 4-manifold bounded by
$(Y,\mathfrak{s})$ and $g_{W}$ be a metric of $W$ extending $g$. Then the
index of the Dirac operator $\mathcal{D}_{W}(\mathfrak{s}_{W})$ of
$(W,\mathfrak{s}_{W})$ associated with $g_{W}$ and the signature of $W$ are
described as

$\displaystyle\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})$
$\displaystyle=-\frac{1}{24}\int_{W}p_{1}-\frac{1}{2}(\dim\ker\mathcal{D}_{Y}(s)+\eta^{\mathrm{Dir}}(Y,s,g)),$
$\displaystyle\sigma(W)$
$\displaystyle=\frac{1}{3}\int_{W}p_{1}-\eta^{\mathrm{sign}}(Y,g).$

It follows that

$(1)\quad
n(Y,\mathfrak{s},g)=-\frac{1}{8}(4\dim\ker\mathcal{D}_{Y}(\mathfrak{s})+4\eta^{\mathrm{Dir}}(Y,\mathfrak{s},g)+\eta^{\mathrm{sign}}(Y,g)).$

If $g$ is a metric of positive scalar curvature,
$\dim\ker\mathcal{D}_{Y}(\mathfrak{s})=0$. For a spherical 3-manifold with
spin structure $(S,\mathfrak{s})$ equipped with the standard metric $g$, we
have the following equation as a special case of Proposition 3, which is
deduced from Proposition 1 and Proposition 2.

$(2)\quad\overline{\mu}(S,\mathfrak{s})=w(S,cS,\mathfrak{s}_{cS})=-\frac{1}{8}(4\eta^{\mathrm{Dir}}(S,\mathfrak{s},g)+\eta^{\mathrm{sign}}(S,g))$

where $\mathfrak{s}_{cS}$ is a spin structure on the cone $cS$ over $S$
extending $\mathfrak{s}$. It follows from (1), (2) and Proposition 6 that
$\kappa(S,\mathfrak{s})=-\overline{\mu}(S,\mathfrak{s})$.

######  Remark 4.

To describe the above equation we assume that the Clifford multiplication $c$
of the volume form $\mathrm{vol}_{Y}$ of $Y$ satisfies
$c(\mathrm{vol}_{Y})=-1$ as in [17]. If we define $c$ so that it satisfies
$c(\mathrm{vol}_{Y})=1$ as in [11], [13], [14], the sign of
$\eta^{\mathrm{Dir}}(Y,\mathfrak{s},g)$ should be changed and the first
equation in Proposition 2 should be replaced with
$\eta^{\mathrm{Dir}}(S,\mathfrak{s},g)=-2\delta^{\mathrm{Dir}}(S,\mathfrak{s})$.

Suppose that a rational homology 3-sphere with spin structure
$(Y,\mathfrak{s})$ bounds a compact spin 4-orbifold $(X,\mathfrak{s}_{X})$
with $\mathfrak{s}_{X}|_{Y}=\mathfrak{s}$. Let $(S_{i},\mathfrak{s}_{i})$
$(i=1,\dots,n)$ be spherical 3-manifolds, which are the links of all isolated
singularities of $X$, where a spin structure $\mathfrak{s}_{i}$ is induced
from $\mathfrak{s}_{X}$. Putting
$X_{0}=X\setminus\cup_{i=1}^{n}\operatorname{Int}cS_{i}$, we have a compact
spin 4-manifold with
$\partial(X_{0},\mathfrak{s}_{X_{0}})=(Y,\mathfrak{s})\cup\cup_{i=1}^{n}(-S_{i},\mathfrak{s}_{i})$
whose spin structure $\mathfrak{s}_{X_{0}}$ is induced from
$\mathfrak{s}_{X}$. Furthermore $(X_{0},\mathfrak{s}_{X_{0}})$ contains a spin
submanifold $W$ of codimension $0$ with $\partial
W=\sharp_{i=1}^{n}S_{i}\cup\cup_{i=1}^{n}(-S_{i})$ which is constructed from
the collars of $S_{i}$ by attaching 1-handles in $X_{0}$. Let
$X_{0}^{\prime}=X_{0}\setminus\operatorname{Int}W$ and
$\mathfrak{s}_{X_{0}^{\prime}}$ be a spin structure induced from
$\mathfrak{s}_{X}$. Then $(X_{0}^{\prime},\mathfrak{s}_{X_{0}^{\prime}})$ is a
spin cobordism from
$(S_{0},\mathfrak{s}_{0})=(\sharp_{i=1}^{n}S_{i},\sharp_{i=1}^{n}\mathfrak{s}_{i})$
to $(Y,\mathfrak{s})$. We note that $S_{0}$ admits a metric $g_{0}$ of
positive scalar curvature ([7], [20]). Thus if we choose a metric $g_{W}$ on
$W$ extending $g_{0}$ and the standard metrics $g_{i}$ of $S_{i}$ $(i\geq 1)$,
we obtain a $G$ equivariant map $f:Z_{0}\to Z_{1}$ for the cobordism $W$ from
$\cup_{i=1}^{n}S_{i}$ to $S_{0}$ of the following form (up to stabilization)
by a procedure similar to those in [13], [14] since the Conley index for a
disjoint union of $S_{i}$ is the smash product of the Conley indices of
$S_{i}$ ([13]).

$\displaystyle
Z_{0}=(\mathbf{H}^{N+\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})-\sum_{i=1}^{n}\dim_{\mathbf{H}}(V_{i})^{0}_{-\nu}(\mathbf{H})})^{+}\wedge(\widetilde{\mathbf{R}}^{M-\sum_{i=1}^{n}\dim_{\mathbf{R}}(V_{i})^{0}_{-\nu}(\mathbf{R})})^{+}\wedge(\wedge_{i=1}^{n}I_{i})$
$\displaystyle
Z_{1}=(\mathbf{H}^{N-\dim_{\mathbf{H}}(V_{0})^{0}_{-\nu}(\mathbf{H})})^{+}\wedge(\widetilde{\mathbf{R}}^{M-\dim_{\mathbf{R}}(V_{0})^{0}_{-\nu}(\widetilde{\mathbf{R}})+b_{2}^{+}(W)})^{+}\wedge
I_{0}$

where $(V_{i})^{0}_{-\nu}$ and $I_{i}$ are the vector space and the Conley
index for $S_{i}$ defined as above $(0\leq i\leq n)$. Since
$I_{i}=(V_{i})^{0}_{-\nu}$ and $b_{2}^{+}(W)=0$, $Z_{0}$ and $Z_{1}$ are of
type SWF at level $M$ (which can be assumed to be even),
$f^{S^{1}}:Z_{0}^{S^{1}}\to Z_{1}^{S^{1}}$ is a $G$ homotopy equivalence.
Furthermore
$k(Z_{0})=N+\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})$
and $k(Z_{1})=N$. Hence applying Lemma 1 to $f$ we deduce that
$\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})\leq 0$. We
note that since $\sigma(W)=0$, we have

$\displaystyle\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})=-\frac{1}{24}\int_{W}p_{1}-\frac{1}{2}(\eta^{\mathrm{Dir}}(S_{0},\mathrm{s}_{0},g_{0})+\sum_{i=1}^{n}\eta^{\mathrm{Dir}}(-S_{i},\mathfrak{s}_{i},g_{i}))$
$\displaystyle=-\frac{1}{8}(4\eta^{\mathrm{Dir}}(S_{0},\mathfrak{s}_{0},g_{0})+\eta^{\mathrm{sign}}(S_{0},g_{0})+\sum_{i=1}^{n}(\eta^{\mathrm{Dir}}(-S_{i},\mathfrak{s}_{i},g_{i})+\eta^{\mathrm{sign}}(-S_{i},g_{i})))$
$\displaystyle=n(S_{0},\mathfrak{s}_{0},g_{0})-\sum_{i=1}^{n}n(S_{i},\mathfrak{s}_{i},g_{i})$

By a similar map defined for a cobordism $-W$ from $S_{0}$ to
$\cup_{i=1}^{n}S_{i}$, we have
$\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{(-W)}(\mathfrak{s}_{W})=-\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})\leq
0$ since $b_{2}^{+}(-W)=b_{2}^{-}(W)=0$. It follows that
$\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{W}(\mathfrak{s}_{W})=2\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{W}(\mathfrak{s}_{W})=0$
and

$(3)\quad\kappa(S_{0},\mathfrak{s}_{0})=-n(S_{0},\mathfrak{s}_{0},g_{0})=-\sum_{i=1}^{n}n(S_{i},\mathfrak{s}_{i},g_{i})=-\sum_{i=1}^{n}\overline{\mu}(S_{i},\mathfrak{s}_{i}).$

Hereafter we assume that $(Y,\mathfrak{s})$ is a Seifert rational homology
3-sphere with spin structure and choose the spin 4-orbifolds
$(X^{\pm},\mathfrak{s}_{X^{\pm}})$ bounded by $(Y,\mathfrak{s})$ satisfying
the conditions in Proposition 3. Let $(S_{i}^{\pm},\mathfrak{s}_{i}^{\pm})$
$(i=1,\dots,n^{\pm})$ be the spherical 3-manifolds which are the links of the
isolated singularities of $X^{\pm}$, where $\mathfrak{s}_{i}^{\pm}$ is induced
from $\mathfrak{s}_{X^{\pm}}$. Subtracting all the cones $cS_{i}^{\pm}$ and
the spin 4-manifold of codimension $0$, which is a spin cobordism from
$\cup_{i=1}^{n^{\pm}}(S_{i}^{\pm},\mathfrak{s}_{i}^{\pm})$ to
$(S_{0}^{\pm},\mathfrak{s}_{0}^{\pm})=(\sharp_{i=1}^{n^{\pm}}S_{i}^{\pm},\sharp_{i=1}^{n^{\pm}}\mathfrak{s}_{i}^{\pm})$
constructed as above from $X^{\pm}$, we obtain compact spin cobordisms
$(X_{0}^{\pm},\mathfrak{s}_{X_{0}^{\pm}})$ from
$(S_{0}^{\pm},\mathfrak{s}_{0}^{\pm})$ to $(Y,\mathfrak{s})$ satisfying

$b_{1}(X_{0}^{\pm})=b_{1}(X^{\pm})=0,\ \
b_{2}^{+}(X_{0}^{+})=b_{2}^{+}(X^{+})\leq 1,\ \
b_{2}^{-}(X_{0}^{-})=b_{2}^{-}(X^{-})\leq 1.$

Since $(S_{0}^{\pm},\mathfrak{s}_{0}^{\pm})$ is Floer $K_{G}$ split, we have
the following inequalities by applying Theorem 4 and Theorem 5.

$\displaystyle(4)\quad-b_{2}^{-}(X_{0}^{\pm})+\frac{\sigma(X_{0}^{\pm})}{8}-1$
$\displaystyle\leq\kappa(S_{0}^{\pm},\mathfrak{s}_{0}^{\pm})-\kappa(Y,\mathfrak{s})$
$\displaystyle\leq b_{2}^{+}(X_{0}^{\pm})+\frac{\sigma(X_{0}^{\pm})}{8}-1$

unless $b_{2}^{+}(X_{0}^{\pm})=0$. If $b_{2}^{+}(X_{0}^{+})=0$, we replace
$X_{0}^{+}$ with $X_{0}^{+}\sharp S^{2}\times S^{2}$ to obtain the inequality
$\kappa(S^{+},\mathfrak{s}_{S^{+}})-\kappa(Y,\mathfrak{s})\leq\sigma(X_{0}^{+})/8$
by applying Theorem 5. If $b_{2}^{-}(X_{0}^{-})$ is even (in fact $0$), then
the first inequality of (4) for $S^{-}$ can be replaced with

$(5)\quad-
b_{2}^{-}(X_{0}^{-})+\frac{\sigma(X_{0}^{-})}{8}\leq\kappa(S^{-},\mathfrak{s}_{S^{-}})-\kappa(Y,\mathfrak{s}).$

On the other hand it follows from Propositions 1, 2, 3 and the equations (1),
(2), (3) that

$\displaystyle(6)\quad$
$\displaystyle\overline{\mu}(Y,\mathfrak{s})=w(Y,X^{\pm},\mathfrak{s}_{X^{\pm}})$
$\displaystyle=\frac{1}{8}(\sigma(X_{0}^{\pm})-\sum_{i=1}^{n}(4\eta^{\mathrm{Dir}}(S_{i}^{\pm},\mathfrak{s}_{i}^{\pm},g_{i}^{\pm})+\eta^{\mathrm{sign}}(S_{i}^{\pm},g_{i}^{\pm})))$
$\displaystyle=\frac{1}{8}\sigma(X_{0}^{\pm})+\sum_{i=1}^{n}n(S_{i}^{\pm},\mathfrak{s}_{i}^{\pm},g_{i}^{\pm})=\frac{1}{8}\sigma(X_{0}^{\pm})-\kappa(S_{0}^{\pm},\mathfrak{s}_{0}^{\pm})$

Therefore by subtracting (6) from (4), we obtain

$(7)\quad-2\leq-
b_{2}^{-}(X_{0}^{-})-1\leq-\kappa(Y,\mathfrak{s})-\overline{\mu}(Y,\mathfrak{s})\leq
b_{2}^{+}(X_{0}^{+})-1\leq 0$

if $b_{2}^{+}(X_{0}^{+})=1$. We replace $X_{0}^{+}$ with $X_{0}^{+}\sharp
S^{2}\times S^{2}$ or $X_{0}^{+}\sharp 2S^{2}\times S^{2}$ if
$b_{2}^{+}(X_{0}^{+})=0$ to obtain the last inequality by applying Theorem 5.
If $Y$ is a Seifert integral homology 3-sphere (with the unique spin
structure), $\kappa(Y)+\overline{\mu}(Y)=0$ or $2$ since both $\kappa(Y)$ and
$\overline{\mu}(Y)$ are integers and the integral lifts of the Rokhlin
invariant.

Suppose that a Seifert rational homology 3-sphere $(Y,\mathfrak{s})$ is Floer
$K_{G}$ split. Then by Theorem 5 we obtain

$0\leq-
b_{2}^{-}(X_{0}^{-})+1\leq-\kappa(Y,\mathfrak{s})-\overline{\mu}(Y,\mathfrak{s})$

if $b_{2}^{-}(X_{0}^{-})=1$ in addition to (7). We obtain the same estimate by
(5) or applying Theorem 5 to $X_{0}^{-}\sharp S^{2}\times S^{2}$ if
$b_{2}^{-}(X_{0}^{-})=0$. Thus we have
$\kappa(Y,\mathfrak{s})+\overline{\mu}(Y,\mathfrak{s})=0$.

For a general rational homology 3-sphere with spin structure
$(Y,\mathfrak{s})$, it is proved that
$\kappa(Y,\mathfrak{s})+\kappa(-Y,\mathfrak{s})\geq 0$ ([14]). Note that
$\kappa(Y,\mathfrak{s})$ is not necessarily the same as
$-\kappa(-Y,\mathfrak{s})$, while we have
$\overline{\mu}(-Y,\mathfrak{s})=-\overline{\mu}(Y,\mathfrak{s})$. If $Y$ is a
Seifert rational homology 3-sphere, the above result shows that
$0\leq\kappa(Y,\mathfrak{s})+\overline{\mu}(Y,\mathfrak{s})\leq 2$,
$0\leq\kappa(-Y,\mathfrak{s})-\overline{\mu}(Y,\mathfrak{s})\leq 2$, and hence
$0\leq\kappa(Y,\mathfrak{s})+\kappa(-Y,\mathfrak{s})\leq 4$. If $Y$ is a
Seifert integral homology 3-sphere, we also have
$\kappa(-Y)+\overline{\mu}(-Y)=\kappa(-Y)-\overline{\mu}(Y)=0$ or $2$. It
follows that $\kappa(Y)+\kappa(-Y)=0$,$2$, or $4$. If $Y$ has a fibration such
that one of the multiplicities is even and has a positive degree, then we
apply the above inequalities for $Y$ to $X^{-}=X^{+}$ satisfying
$b_{2}^{-}(X^{\pm})=0$ and $b_{2}^{+}(X^{\pm})=1$, which is chosen in
Proposition 3. It follows from (5), (6), (7) that
$\kappa(Y,\mathfrak{s})=-\overline{\mu}(Y,\mathfrak{s})$.

This completes the proof of Theorem 2.

######  Example 1.

We give a list of the values of $\kappa$, $\overline{\mu}$ together with
$\beta$ and $\underline{d}$ of some Brieskorn homology 3-spheres. The
computations of the $\kappa$ invariants below are due to Manolescu [14]. It is
pointed out in [14] that $\pm\Sigma(2,3,12n+1)$ and $\pm\Sigma(2,3,12n+5)$ are
Floer $K_{G}$ split, while $\pm\Sigma(2,3,12n-1)$ and $\pm\Sigma(2,3,12n-5)$
are not.

|  $\kappa$ |  $\overline{\mu}$ |  $\beta$ |  $\underline{d}$
---|---|---|---|---
$\ \Sigma(2,3,12n-1)$ |  2 |  0 |  0 |  0
$-\Sigma(2,3,12n-1)$ |  0 |  0 |  0 |  0
$\ \Sigma(2,3,12n+1)$ |  0 |  0 |  0 |  0
$-\Sigma(2,3,12n+1)$ |  0 |  0 |  0 |  0
$\ \Sigma(2,3,12n-5)$ |  1 |  1 |  -1 |  -2
$-\Sigma(2,3,12n-5)$ |  1 |  -1 |  1 |  2
$\ \Sigma(2,3,12n+5)$ |  1 |  -1 |  1 |  2
$-\Sigma(2,3,12n+5)$ |  -1 |  1 |  -1 |  -2

######  Remark 5.

If $Y$ is a Seifert fibration of the form
$\\{b,(a_{1},b_{1}),\dots,(a_{n},b_{n})\\}$, then $Y$ is represented by a
framed link $\cup_{i=0}^{n}K_{i}$, where $K_{0}$ is an unknot with framing $b$
and $K_{i}$ $(i\geq 1)$ is a meridian of $K_{0}$ with framing $-b_{i}/a_{i}$.
Let $h$ be a meridian of $K_{0}$ (a general fiber of $Y$) and $g_{i}$ be a
meridian of $K_{i}$ $(i\geq 1)$. Then a spin structure $\mathfrak{s}$ on $Y$
corresponds to a homomorphism
$c:H_{1}(S^{3}\setminus\cup_{i=0}^{n}K_{i},\mathbf{Z})\to\mathbf{Z}_{2}$
satisfying

$\displaystyle a_{i}c(g_{i})+b_{i}c(h)\equiv a_{i}b_{i}\pmod{2},$
$\displaystyle\sum_{i=1}^{n}c(g_{i})+bc(h)\equiv b\pmod{2}$

([25]. The description of the Seifert invariants in [25] is different from
that of this paper, but the above condition is still valid.) Suppose that all
$a_{i}$ are odd and $\sum_{i=1}^{n}b_{i}\equiv b\pmod{2}$ (such a case only
occurs when $|H_{1}(Y,\mathbf{Z})|$ is even). Then we have a spin structure
$\mathfrak{s}$ on $Y$ corresponding to $c$ that satisfies $c(h)\equiv 0$,
$c(g_{i})\equiv b_{i}$ $(i\geq 1)$. In this case $X^{\pm}$ in Proposition 3
can be also chosen so that $X^{+}=X^{-}$ satisfying $b_{2}^{+}(X^{\pm})=1$ and
$b_{2}^{-}(X^{\pm})=0$ if $\deg Y>0$, and $b_{2}^{-}(X^{\pm})=1$ and
$b_{2}^{+}(X^{\pm})=0$ if $\deg Y<0$ ([25]). Thus it follows from the proof of
Theorem 2 that $\kappa(Y,\mathfrak{s})=-\overline{\mu}(Y,\mathfrak{s})$ if
$\deg Y>0$.

######  Remark 6.

If a Seifert rational homology 3-sphere with spin structure $(Y,\mathfrak{s})$
bounds a negative definite spin 4-manifold $W$, the inequalities in Theorem 1
coincide with those in [26]. Furthermore for a Seifert integral homology
3-sphere, the second inequality in Theorem 1 coincides with those in terms of
the $\beta$ invariant ([13]), and in terms of the $\underline{d}$ invariant
([8]) under the relations $\beta(Y)=-\overline{\mu}(Y)$ ([2], [23]) and
$\underline{d}(Y)=-2\overline{\mu}(Y)$ ([3]).

######  Remark 7.

We have $\kappa(Y)+\overline{\mu}(Y)=0$ or $2$ for a Seifert integral homology
3-sphere $Y$ by Theorem 2. If $\kappa(Y)=-\overline{\mu}(Y)$, then Corollary 1
shows that $\overline{\mu}(Y)\leq b_{2}^{+}(W)+\frac{\sigma(W)}{8}-1$ for a
compact spin 4-manifold $W$ bounded by $Y$, which give the better estimate
than Theorem 1. If $\kappa(Y)=2-\overline{\mu}(Y)$, the estimate in Theorem 1
is slightly better than or the same as that of $\overline{\mu}(Y)$ given in
Corollary 1.

J.Lin [9] proves other constraints on the intersection forms of spin
4-manifolds bounded by homology 3-spheres in terms of $\mathrm{KO}$
invariants, which give estimates better than those in Theorem 1 and Theorems
4, 5 in some cases. In [9] several estimates for spin 4-manifolds bounded by
certain Brieskorn homology 3-spheres are also given by considering the closed
spin 4-orbifolds as in [4]. We obtain similar estimates by using Proposition 3
as follows. Suppose that a Seifert rational homology 3-sphere $Y$ (with spin
structure) bounds a spin 4-manifold $W$. Let $Z=X^{-}\cup(-W)$ be a closed
spin 4-orbifold, where $X^{-}$ is a spin 4-orbifold bounded by $Y$ chosen in
Proposition 3. Then we have a $G$ equivariant map

$f:(\mathbf{H}^{\operatorname{ind}_{\mathbf{H}}\mathcal{D}_{-Z}})^{+}\to(\widetilde{\mathbf{R}}^{b_{2}^{+}(-Z)})^{+}$

whose restriction to the $G$-fixed point set is a homotopy equivalence (we
choose $Z$ according to our sign convention). It is proved in [9] that there
is no such map of the form
$f:(\mathbf{H}^{4\ell})^{+}\to(\widetilde{\mathbf{R}}^{8\ell+2})^{+}$ if
$\ell>0$. Note that $b^{-}(X^{-})\leq 1$ and
$\operatorname{ind}_{\mathbf{C}}\mathcal{D}_{-Z}=\overline{\mu}(Y)-\frac{\sigma(W)}{8}$.
It follows that if $\overline{\mu}(Y)-\frac{\sigma(W)}{8}>0$ and divisible by
8, then

$\overline{\mu}(Y)\leq b_{2}^{+}(W)+\frac{\sigma(W)}{8}-2,$

which gives a better estimate than those by Theorem 1 or Corollary 1.

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